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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I ( L 31 I >.■ A MANUAL OF SPHERICAL AND PEACTICAL ASTRONOMY: BMBftACINO THE GENERAL PROBLEMS OF SPHERICAL ASTRONOMY, THE SPECIAL APPLICATIONS TO NAUTICAL ASTRONOMY, AND THE THEORY AND USE OF FIXED AND PORTABLE ASTRO- NOMICAL INSTRUMENTS. WITH AN APPENDIX ON THE METHOD OF LEAST SQUARES. BY WILLIAM CHAUVENET, nonssoK or lUTBiMAncB and ibtboitomt or wasbihotos usmiaiTT, baxht louu. VOL. I. SPHEKICAL ASTRONOMY. SBCONB EDITION, BEVISED AND CORRECTED. PHILADELPHIA: J. B. LIPPINCOTT & CO. LONDON: TRttBNER & CO. 1864. Enterwl, •cconling to Act of Ci^ngnm, in the year 1863, by J. B. LIPnXCUTT A CX). Tn the Clerk'ii OAre of the Dintrlrt Court ttt the United SUtee for the EMtem DiHtrict uf PeoDMylvaoU. PREFACK The methods of investigation adopted in this work are in accordance with what may be called the modem school of practical astronomy, or more distinctively the Ger- man school, at the head of which stands the unrivalled Bessel. In this school, the investigations both of the general problems of Spherical Astronomy and of the Theory of Astronomical Instruments are distinguished by the gene- rality of their form and their mathematical rigor. When approximative methods are employed for convenience in practice, their degree of accuracy is carefully determined by means of exact formulaB previously investigated ; the latter being developed in converging series, and only such terms of these series being neglected as can be shown to be insen- sible in the cases to which the formulaB are to be applied. And it is an essential condition of all the methods of com- putation from data furnished by observation, that the errors of the computation shall always be practically insensible in relation to the errors of observation : so that our results shall be purely the legitimate deductions from the observa- tions, and free from all avoidable error. It is another characteristic feature of modem spherical astronomy, that the final formulae furnished to the practical computer are so presented as seldom to require accompany- ing verbal precepts to distinguish the species of the unknown angles and arcs ; and this results, in a great measure, from the consideration of the general spherical triangle^ or that in which the six parts of the triangle are not subjected to the 3 PREFACE. condition that they shall each be less than 180°, but may have any values less than 360°, all ambiguity as to their species being removed by determining them, when necessary, by two of their trigonometric functions, usually the sine and the cosine. This feature is mainly due to Gauss, and was prominently exhibited in his Theoria Motus Corporum Ckx- lestiuniy published in 1809. The English and American astronomers have been slow to adopt this manifest improve- ment ; in evidence of which I may remark that the general spherical triangle was not treated of in any work in the English language, so far a^ I know, prior to the publication of my Treatise on Plane and Spherical Trigonometry^ in the year 1850. In the present work, I assume the reader to be acquainted with this form of spherical trigonometry, and to accept its fundamental equations in their utmost generality. A third and eminently characteristic feature of modem astronomy, is the use which it makes, in all its departments, of the method of least squares, namely, that method of combining observations which shall give the moat pnjhaUe results, or which shall be exposed to the least probable errors. This method is also due to Gauss, who (though anticipated in the publication of one of its practical rules by Legendre) was the first to give a philosophical exposition of its princi- ples. The direct effect of this improvement is not only that the most probable result in each case is obtained, but also that the relative degree of accuracy of that result is deter- mined, and thus the degixje of confidence with which it may be received imd the weight which it may be allowed to have in subsequent discussions. Judiciously employed, it ser\'es to indicate when a particular process has reached the limit of accuracy which it can afford, thereby saves fruitless labor, directs inquiry into new channels, and contributes greatly to accelerate the progress of the science. Whilst the scnence has been rapidly advancing in Europe, wo have in this country not been idle. Two of the most important improvements in practical astronomy have had PREFACE. 5 their origin in the United States,— the method of finding differences of geographical longitude by the electric telegraph, and that of finding the geographical latitude by the zenith telescope. These are the direct offspring of our admirably conducted Coast Survey, which, with the aid of these methods, both of the greatest simplicity, has fixed the lati- tudes and relative longitudes of a series of points on our coast with a degree of accuracy wholly unapproached in any previous work of this kind. This extreme accuracy will be apparent to the reader who will refer to the examples here given, which have been selected (almost at random) from the records of the Survey. It is perhaps necessary to say a few words here respect- ing those portions of this treatise in which I have ventured to substitute my own methods for those heretofore employed. My method of reducing lunar distances, which was first published in the American Ephemeris for 1855, is here re- produced, together with the necessary tables for its applica- tion. But I have first, for the sake of completeness, given the usual rigorous solution, although this is confessedly too laborious for ordinary use, and especially for use at sea. The approximative methods heretofore proposed may be divided into two classes : first, those based upon sufficiently precise formulsB, but such that the tables required in their applica- tion are adapted only to a mean state of the atmosphere; and second, those based upon incomplete formulsa. As to the first class, the trouble of correcting the tabular numbers for the barometer and thermometer would render the methods as laborious as the rigorous method, and it is therefore the usual practice, at sea, to disregard these correc- tions altogether, thus introducing a greater error than would follow from the use of the more incomplete formulaa of the second class, if in the latter these corrections were taken into account. But, as to the methods of the second class (of which there are several in common use), it will be found upon examination that the omitted terms of the formulaa are not so small as to be insensible even in relation to the rather large errors of observation which are unavoidable ip the use of the sextant. The defects of both classes are supposed to be avoided in my new method ; for, first, 1 have deduced a rigorous formula from which is derived an ap- proximate one, practically perfect, representmg the true cor- rection of the lunar distJinee within one second of arc in every cose that can occur in practice; and, second, I have arranged this formula so that it not only requires extremely simple tables in its application, but also tfie Utfmlar mtiiiJitrH retjuire iio correction for the Luromcter and thermometer, the corrections for the state of these instruments being intro- duced in a simple manner in fonning the anjumentu of the tables. In applying this method with It^arithms of only four decimal places, the true distance is usually obtained within less than two seconds of arc, a degree of accuracy fiu" greater than is necessary in relation to our present means of observing the distance. It is, in fact, quite as accurate m practice as Bessel's theoretically exact method when tlie latter is also carried out with four<place lugarithiits. I think, therefore, tinit I may justly prefer my own method not only to the imperfect approximative methods atwve referred to, but also tt) Bessel's method, which n-quires an extended Ephemeris wholly different from that now in use, and iti withal more lalmrious. The Gaussian method of reducuig circummcridian alti- tudi'H of the sun by n-ferring them lo the instant of the auw's maximum altitude, is in thi.s work rigorously investi- gated, and a small t*'nn, overlooked or disregarded by Gauss, ]iaf( been added to the formula. A new and brief approximative method of finding the latitude by two altitudes near tlio meridian wlien the time is not known, is given in Vol. 1. Arts. 105 and 204, and another by thn« altitudes near tlie meridian, in Art- 205, which will probably 1k^ found useful as nautiejd nietb()ds. The subject of Eclipses will be fomid treated with more PREFACE. 7 than usual completeness. The fundamental formulaB adopted are those of Bessel's theory, but the solutions of the various problems relating to the prediction of solar eclipses for the earth generally are mostly new. The rigorous solutions of these problems given by Bessel in his Analyse der Finster- nesse are not required for the usual purposes of prediction, however interesting they may be as specimens of refined and elegant analysis. On the other hand, the approximate solutions commoDly given appear to be unnecessarily rude. Those that I have substituted will be found to be very little if at all more laborious than the latter, while they are almost as precise as the former, and by a very little additional labor (that Is, by repeating only some parts of the computation for a second or third approximation) may be rendered quite exact. So far as I can find, no one has heretofore treated distinct- ively of the occultations of planets by the moon, and these phenomena have been dismissed as simple cases of the general theory of eclipses, in which both the occulting and the occulted body are spherical. But in almost every oc- cultation of one of the principal planets, the planet will be either a spheroidal body fully or partially illuminated by the sun, or a spherical body partially illuminated : so that, in the general case, we have to consider the disc of the oc- culted body as bounded by an ellipse or by two different semi-eUipses. I have discussed this general case at length, and have adapted the theory to each planet specially. The additional computations required to take into account the true figure of the planet's disc are sufficiently brief and simple. The case of the occultation of a cusp of Venus or Mercury is included in the discussion, and also the occulta- tion of Saturn's rings. The well known formula for predicting the transits of the inferior planets over the sun's disc, first given by Lagrange, is here rendered more accurate by introducing a considera- tion of the compression of the earth ; and a uew and simple demonstration of the formula is given. In the practical portions of the work, and especially in the second volume, I have endeavored to give every import- ant precept for the guidance of observers, deduced from the labors of others or suggested by my own experience. All the principal methods are illustrated by examples from actual observation. I have taken especial pains throughout the work to ex- hibit the mode of discussing the probable errors of the results obtained by observations, and have given numerous examples of the application of tlie method of least squares. This method is applicable in almost all the physical sciences where numerical results are to ]»e deduced, and, therefore, does not necessarily form a part of a work on astronomy ; but, as I could not refer my reader to any work in the English language for a sufficient account of the method, I have prepared a concise treatise upon it, which forms the Appendix. In this, I have confined myself chiefly to the parts of the theory required in practical astronomy, and have endeavored to present its principles in a simple yet rigonius manner (so far as the subject allows), taking as a ba-sis known theorems of the calculus of probabilitiesj and follow- ing principally the processes first proposed by Gauss. In this Ap|>endix I have treated of Peirck's Criterion for the rejection of doubtful oljservatious, which is already well known to American aj^tronomers, and is now constantly applied in the discussion of observations upon our Coast Survey. Objectittns have Ijeen made to the criterion, l>ut none that would not apply equally well to the method of least stjuurcs itself. To tliu^* who have not been able to follow pErncE's investigation, the simple approximate cri- t4>rion which I have suggested at the end of the Ap{)endix may prove acceptable. It is derived directly from the fim- dauiental formula of the method of least stjuorcs, and leads PRBFACE. 9 to the rejection of nearly the same observations as that of Peirce. The plates at the end of the work exhibit in minute detail the instruments now chiefly employed by astronomers. To have given more, with the necessary explanations, would have led me too far into the mere history of the subject, and would have occupied space which I thought it preferable to fill with discussions relating to the leading instruments now in use. The scale of these plates is purposely made quite small; but the great precision with which they are executed will enable the reader to measure from them the dimensions of all the important parts of each of the principal instru- ments. I am greatly indebted for the perfection of these drawings to the engravers, the Messrs. Illman Brothers, of Philadelphia. Such auxiliary tables as seemed to be necessary to the reader in using these volumes have been given at the end of Vol. II. Some of these are new. Most of those which have been derived from other sources have been either re- computed or tested by differences and corrected. To insure their accuracy, they have also been tested by differences after being in type. For the very complete index to the whole work, I am indebted to my friend. Prof J. D. Crehore, of Washington University. In conclusion, I desire to express my obligations to those citizens of Saint Louis who, without solicitation, have gene- rously assumed a share of the risk of publication. Their liberal spirit has been met by a corresponding liberality on the part of my publishers, who have spared no expense in the typographical execution. I shall be content if their expectations are not wholly disappointed, and the work contributes in any degree to the advancement of the noblest of the physical sciences. Washington Unitebsitt, Saint Louis, January 1, 1863. CONTENTS OF VOL. L SPHERICAL ASTRONOMY. CHAPTER I. PAOl The Celestial Sphbrb — Spherical and Rectangular Co-obdinates 17 Spherical co-ordinates 18 Transformation of spherical co-ordinates 27 Rectangular co-ordinates. .. 43 Transformation of rectangular co-ordinates 48 Differential yariations of co-ordinates 50 CHAPTER II. Time — Use of the Ephemeris — Interpolation — Star Catalogues 62 Solar time 68 Sidereal time 59 Hour angles 64 Ephemeris 68 Interpolation by differences of any order 79 Star catalogues 91 CHAPTER III. Figure and Dimensions of the Earth 96 Reduction of latitude 97 Radius of the terrestrial spheroid for given latitudes 99 Normal, &c 101 CHAPTER IV. Reduction of Obsbrtations to the Centre of the Earth 108 Parallax 104 Refraction. — General laws of refraction 127 Tables of refraction 180 Differential equation of the atmospheric refraction 186 Integration of the differential equation with Bououer's hypothesis 186 Integration with Bessel's hypothesis according to the methods of Krahp and Laplace 148 11 CONTENTS. TAIOM Coiutruction of Bbssbl's Table 165 Refraction in right ascension and declination 171 Dip of the horiion 172 Semidiameters of celestial bodies „ 180 Augmentation of the moon's semidiameter 183 Contraction of the sun's and the moon's semidiameters by refWu:tion 184 Reduction of obserred zenith distances to the centre of the earth 189 CHAPTER V. FiXDiNO THE Time by Asteonomical Observations 193 1st Method.— Bj transits 1% 2d Method. — Bj equal altitudes 190 dd Method. — By a single altitude or zenith distance 200 Correction for second differences of zenith distance 213 4th Method. — By the disappearance of a star behind a terrestrial object.... 217 Time of rising and setting of the stars 218 FlMDING THE TiMB AT 8bA 219 Ist Method. — By a single altitude 219 2d Method.— By equal altitudes 220 CHAPTER VI. FiXDiBO THE Latitude bt Astbonomical Obsbbvations 223 1st Method. — By meridian altitudes or zenith distances....... 223 Combination of pairs of stars whose meridian zenith distances are nearly equal (see Vol. II., Zenith Telescope) 226 Meridian altitudes of a circumpolar star 226 Meridian zenith distances of the sun near the solstices 228 2d Method. — By a single altitude at a given time 229 8d Method. — By reduction to the meridian when the time is given 238 Circummeridian altitudes 235 Gauss's method of reducing circummeridian altitudes of the sun 244 Limits of the redaction to the meridian 261 4ih Method.— By the Pole Star 263 6th Method. — By two altitudes of the same star, or different stars, and the elapsed time between the observations.. 267 General solution 268 Caillet's formulsD for a fixed star or the sun 264 Correetionof this method for the sun 266 6th Method. — By two altitudes of the same or different stars, with the difference of their azimuths .« 277 7Ui Method. — By two different stars observed at the same altitude, when the time is given 277 At nearly the same altitude, observed with the zenith telescope 279 8th Method. — By three stars observed at the same altitude (Gauss's method) 280 The same by Caoxoli's formulsB 286 By a number of stars observed at the same altitude, treated by the Method of Least Squares 280 9th Method. — By the transits of stars over vertical circlet (see Vol. II., Transit Instrument in the Prime Vertieml) ~ 298 CONTENTS. 10th Method. — By altitudes ne&r the meridian when the time is not known... 296 (A.) By two altitudes near the meridian and the chronometer times of the obserTations, when the rate of the chronometer is known, but not its correction 296 (B.) By three altitudes near the meridian and the chronometer times of the obseryations, when neither the correction nor the rate of the chronometer is known 299 (C.) By two altitudes near the meridian and the difference of the azimuths 801 (D.) By three altitudes near the meridian and the differences of azimuths 802 11th Method. — By the rate of change of altitude near the prime vertical 808 FiNDiMO THB Latitude at Ska 804 1st Method. — By meridian altitudes 804 2d Method. — By reduction to the meridian when the time is given 307 8d Method. — By two altitudes near the meridian when the time is not known 807 4th Method. — By three altitudes near the meridian when the time is not known 809 6th Method. — By a single altitude at a given time... 810 6th Method. — By the change of altitude near the prime vertical 811 7th Method.— By the Pole Star 311 8th Method. — By two altitudes with the elapsed time between them 813 CHAPTER VII. Finding the Longitude bt Astronomical Observations 817 1st Method. — By portable chronometers 317 Chronometrio expeditions 823 2d Method.— By signals 887 Terrestrial signals 387 Celestial signals, (a) Bursting of a meteor. (6) Beginning or end- ing of an eclipse of the moon, (c) Eclipses of Jupiter's satellites. {d) Occultations of Jupiter's satellites, (e) Transits of the satel- lites over Jupiter's disc. (/) Transits of the shadows of the satel- lites over Jupiter's disc, (g) Eclipses of the sun, Occultations of stars and planets by the moon. [See Chapter X.] 389 8d Method. — By the electric telegraph 341 Method of star signals 842 4th Method. — By moon culminations 850 Pbiroe's method of correcting the epbemeris 358 Combination of moon culminations by weights 368 5th Method. — By azimuths of the moon, or transits of the moon and a star over the same vertical circle 871 6th Method. — By altitudes of the moon 882 (A.) — By the moon's absolute altitude 388 (B.) — By equal altitudes of the moon and a star observed with the Zenith Telescope 386 7th Method. — By lunar distances 893 (A.) — Rigorous method 395 (B.) — Approximative method 402 CONTENTS. PAOB FiXDIXO THE LOMQITUDE AT SbA 420 Bj chronometers « 420 By lunar distances ■ 422 Bj the eclipses of Jupiter's satellites ^ 428 Bj the moon*s altitude 428 Bj occultations of stars by the moon 424 CHAPTER VIII. FiXDixo A Ship's Place at Sea bt Circles of Positiox — Sumscer's Method... 424 CHAPTER IX. The Meridulh Lixe axd Variatiox or tub Compass 429 CHAPTER X. Eclipses 486 Solar Eclipses. Prediction for the earth generally 486 Fundamental equations 489 Outline of the shadow 456 Rising and setting limits 466 Cunre of maximum in horizon 475 Northern and southern limits 480 Cunre of central eclipse 491 Limits of total or annular eclipse 498 Prediction for a giTen place 505 Correction for atmospheric refraction in eclipses 515 Correction for the height of the obserrer aboTe the lerel of the sea 517 Application of obserred solar eclipses to the determination of terrestrial longi- tudes and the correction of the elements of the computation 518 Lunar eclipses 542 Occultations of ftxed stars by the moon 549 Terrestrial longitudes from occultations of stars... 550 Prediction of occultations 557 Limiting parallels.. 561 Occultations of planets by the moon 565 Apparent form of a planet's disc 566 Terrestrial longitude from occultations of planets 578 Transits of Venus and Mercury - 591 Determination of the solar parallax 592 Prediction for the earth generally •••• 598 Occultation of a fixed star by a planet 601 CHAPTER XL PRBCBSSIOIC, NvTATIO.^, ABERRATION, AXI> .VxicrAL PaRALLAX OF THE FiXBD Stars 002 Precession ^*>0A Nutation 624 Aberration 628 ParmlUz 643 Mmii RBd appttrent places of stars 645 CONTENTS. CHAPTER XII. PAOI Deterxixatioh of the Obliquity of the Ecliptic and the Absolute Right Ascensions and Declinations of Stars by Observation 658 Obliqaitj of the ecliptic 069 Equinoctial points, and absolute right ascension and declination of the fixed stars 666 CHAPTER XIII. Determination of Astronomical Constants by Observation 671 Constants of refiraotion 671 Constant of solar parallax 678 Constant of lunar parallax 680 Mean semidiameters of the planets 687 Constant of aberration and heliocentric parallax of fixed stars 688 Constant of nutation 698 Constant of precession 701 Motion of the sun in. space 703 SPHERICAL ASTRONOMY. CHAPTER I. THE CELESTIAL SPHERE — SPHERICAL AND BECTANGULAE CO-ORDINATES. 1. From whatever point of space an observer be supposed to view the heavenly bodies, they will appear to him as if situated upon the surface of a sphere of which his eye is the centre. If, without changing his position, he directs his eye successively to the several bodies, he may learn their relative directions, but cannot determine either their distances from himself or from each other. The position of an observer on the surface of the earth is, however, constantly changing, in consequence, 1st, of the diur- nal motion, or the rotation of the earth on its axis ; 2d, of the annual motion, or the motion of the earth in its orbit around the sun. The changes produced by the diurnal motion, in the appa- rent relative positions or directions of the heavenly bodies, are different for observers on different parts of the earth's surface, and can be subjected to computation only by introducing the elements of the observer's position, such as his latitude and longitude. But the changes resulting from the annual motion of the earth, as well as from the proper motions of the celestial bodies themselves, may be separately considered, and the directions of all the known celestial bodies, as they would be seen from the centre of the earth at any given time, may be computed Vol. L— 2 17 18 THE CELESTIAL SPHERE. according to the laws which have been found to govern the motions of these bodies, from data furnished by long series of observations. The complete investigation of these changes and their laws belongs to Physical Astronomy^ and requires the consi- deration of the distances and magnitudes as well as of the direc- tions of the bodies composing the system. Spherical Astronomy treats specially of the directions of the heavenly bodies ; and in this branch, therefore, these bodies are at any given instant regarded as situated upon the surface of a sphere of an indefinite radius described about an assumed centre. It embraces, therefore, not only the problems which arise from the diurnal motion, but also such as arise from the annual motion so far as this aftects the apparent positions of the hea- venly bodies upon the celestial sphere, or their directions from the assumed centre. SPHERICAL CO-ORDINATES. 2. The direction of a point may be expressed by the angles which a line drawn to it from the centre of the sphere, or point of observation, makes with certain fixed lines of reference. But, since such angles are directly measured by arcs on the surface of the sphere, the simplest method is to assign the position in which the point appears when projected upon the surface of the sphere. For this purpose, a great circle of the sphere, supposed to be given in position, is assumed as a primitive circle of refer- ence, and all points of the surface are referred to this circle by a system of secondaries or great circles perpendicular to the primi- tive and, consequently, passing through its poles. The position of a point on the surface will then be expressed by two spherical co-ordinates: namely, 1st, the distance of the point from the pri- mitive circle, measured on a secondary ; 2d, the distance inter- cepted on the primitive between this secondary and some given point of the primitive assumed as the origin of co-ordinates. AVe shall have diflferent systems of co-ordinates, according to the circle adopted as a primitive circle and the point assumed as the origin. 3. First system of co-ordinates. — Altitude aiid azimuth. — ^In this system, the primitive circle is the horizon^ which is that great circle of the sphere whose plane touches the surface of the SPHERICAL CO-ORDINATES. 19 earth at the observer.* The plane of tlie horizon may be con- ceived as that which sensibly coincides with the surface of a fluid at rest. The vertical line is a straight line perpendicular to the plane of the horizon at the observer. It coincides with the direction of the plumb line, or the simple pendulum at rest. The two points in which this line, infinitely produced, meets the sphere, are the zenith and nadir, the first above, the second below the horizon. The zenith and nadir are the poles of the horizon. Secondaries to the horizon are vertical circles. They all pass through the zenith and nadir, and their planes, which are called vertical planes, intersect in the vertical line. Small circles parallel to the horizon are called almucantarSj or parallels of altitude. The celestial meridian is that vertical circle whose plane passes through the axis of the earth and, consequently, coincides with the plane of the terrestrial meridian. The intersection of this plane with the plane of the horizon is the meridian line, and the points in which this line meets the sphere are the north and south points of the horizon, being respectively north and south of the plane of the equator. The prime vertical is the vertical circle which is perpendicular to the meridian. The line in which its plane intersects the plane of the horizon is the east and west line, and the points in which this line meets the sphere are the ea^t and west points of the horizon. The north and south points of the horizon are the poles of the prime vertical, and the east and west points are the poles of the meridian. * In this definition of the horixon we consider the plane tangent to the earth's surface as sensibly coinciding with a parallel plane passed through the centre ; that is, we consider the radius of the celestial sphere to be infinite, and the radius of the earth to be relatirely zero. In general, any number of parallel planes at finite dis- tances must be regarded as marking out upon the infinite sphere the same great circle. Indeed, since in the celestial sphere we consider only direction^ abstracted from dis- tance, all lines or planes haying the same direction — that is, all parallel lines or planes — must be regarded as intersecting the surface of the sphere in the same point or the same great circle. The point of the surface of the sphere in which a number of parallel lines are concciyed to meet is called the vanishing point of those lines ; and, in like manner, the great circle in which a number of parallel planes are conceiTed to meet may be called the vanishing circle of those planes. 20 THE CELESTIAL SPHERE. The altitude of a point of the celestial sphere is its distance from the horizon measured on a vertical circle, and its azimuth is the arc of the horizon intercepted between this vertical circle and any point of the horizon assumed as an origin. The origin from which azimuths are reckoned is arbitrary ; so also is the direction in which they are reckoned; but astronomers usually take the south point of the horizon as the origin, and reckon towards the right hand, from 0° to 360® ; that is, completely around the horizon in the direction expressed by writing the cardinal points of the horizon in the order S.W. N. P]. We may, therefore, also define azimuth as the angle which the vertical i)lane makes with the plane of the meridian. Navigators, however, usually reckon the azimuth from the uorth or south points, according as they are in north or south latitude, and towards the east or west, according as the point of the sphere considered is east or west of the meridian: so that the azimuth never exceeds 180®. Thus, an azimuth which is exjiressed according to the first method simply by 200^ would he expressed by a navigator in north latitude by X. 20® E., and l>y a navigator in south latitude by S. 160® E., the letter prefixed denoting the origin, and the letter aflSxed denoting the direction in which the aziuiuth is reckoned, or whether the point consi- dered is east or west of the meridian. When the point considered is in the horizon, it is often referred to the east or west points, and its distance from the nearest of these points is called its amplitude. Thus, a point in the horizon whose azimuth is 110® is said to have an amplitude of W. 20® X. Since by the diurnal motion the obsen'cr's horizon is made to change its position in the heavens, the co-ordinates, altitude and azimuth, are continually changing. Their values, therefore, will depend not only upon the observer's position on the earth, but upon the time reck(med at his meridian. In>tcad of the altitude of a point, we frequently employ its ze)tith diManee^ which is the arc of the vertical circle between the l»oint and the zenith. The altitude and zenith distance are, therefore, complements of each other. Wc shall hereafter denote altitude by h, zenith distance by ^, azimuth bv -4. AVe shall have then C =^ 90^ — /i /i = 90® — C SPHERICAL CO-ORDIXATES. £1 The value of ^ for a point below the horizon will be greater than 90^, and the corresponding value of /i, found by the for- mula h = 90° — ^, will be negative : so that a negative altitude will express the depression of a point below the horizon. Thus, a depression of 10° \\\\\ be expressed by h = —10°, or ^ = 100°. 4. Senovd system of co-ordinates. — Declination and hour angle, — In this system, the primitive circle is the celestial equator^ or that great circle of the sphere whose plane is perpendicular to the axis of the earth and, consequently, coincides with the plane of the terrestrial equator. This circle is also sometimes called the equinoctird. The diurnal motion of the earth does not change the position of the plane of the equator. The axis of the earth produced to the celestial sphere is called the axis of the heavens: the points in which it meets the sphere are the north and south poles of the equator, or the poles of the heavens. Secondaries to the equator are called circles of declination, and also hour circles. Since the plane of the celestial meridian passes through the axis of the equator, it is also a secondarj^ to the equator, and therefore also a circle of declination. Parallels of declination are small circles parallel to the equator. The declination of a point of the sphere is its distance from the equator measured on a circle of declination, and its hour angle is the angle at either pole between this circle of declination and the meridian. The hour angle is measured hy the arc of the equator intercepted between the circle of declination and the meridian. As the meridian and equator intersect in two points, it is neces- sary to distinguish which of these points is taken as the origin of hour angles, and also to know in what direction the arc which measures the hour angle is reckoned. Astronomers reckon from that point of the equator which is on the meridian above the horizon, towards the west, — that is, in the direction of the apparent diurnal motion of the celestial s])here, — and from 0° to 360°, or from 0* to 24*, allowing 15° to each hour. Of these co-ordinates, the declination is not changed by the diurnal motion, while the hour angle depends only on the time at the meridian of the observer, or (which is the same thing) on the position of his meridian in the celestial sphere. All the observers on the same meridian at the same instant will, for the same st^r, reckon the same declination and hour angle. AVe have 22 THE CELESTIAL SPHERE. thus introduced coordinates of which one is wholly independent of the observer's position and the other is independent of his latitude. AVe shall denote declination by 5, and north declination will be distinguished by prefixing to its numerical value the sign +> and south declination by the sign — . We shall sometimes make use of the polar distance of a point, or its distance from one of the poles of the equator. If we denote it by P, the north polar distance will be found by the formula and the south polar distance by the formula The hour angle will generally be denoted by i. It is to be observed that as the hour angle of a celestial body is continually increasing in consequence of the diurnal motion, it may be con- ceived as having values greater than 360°, or 24*, or greater than any given multiple of 360°. Such an hour angle may be re- gjirdod as expressing the time elai)sed since some given passage of the body over the meridian. But it is usual, when values greater than 360° result from any caleuhition, to deduct 360°. Again, since hour angles reckoned towards the west are always positive, hour-angles reckoned towards the east must have the negative sign : so that an hour angle of 300°, or 20*, may also be expressed by — 60°, or r>. Third system of co-ordinates. — Declination and right ascoision. — In this system, the primitive phme is still the equator, and the first co-ordinate is the same as in the second svstem, namelv, the declination. The second co-ordinate is also measured on the e([uator, but from an origin which is not affected by the diurnal motion. Any i)oint of the celestial ecjuator might be assumed as the origin; but that which is most natunilly indicated is the vernal equinox, to define which some preliminaries are necessarv. The ecliptic is the great circle of the celestial sjihere in which the sun ajjpears to move in consequence of the earth's motion in its orbit. The position of the ecliptic is not absolutely fixed in space; but, according to the definition just given, its i>osition at any instant coincides with that of the great circle in which the SPHERICAL CO-ORDINATES. 23 Buu appears to be moving at that instant. Its annual change is, however, very small, and its daily change altogether insensible. The obliquity of the ecliptic is the angle which it makes with the equator. The points where the ecliptic and equator intersect are called the equinoctial povits, or the equinoxes ; and that diameter of the sphere in which their planes intersect is the line of equinoxes. The vernal equinox is the point through which the sun ascends from the southern to the northern side of the equator ; and the autumnul equinox is that through which the sun descends from the northern to the southern side of the equator. The solstitial points^ or solstices^ are the points of the ecliptic 90° from the equinoxes. They are distinguished as the north- ern and southern, or the summer and winter solstices. Tlie equinoctial colure is the circle of declination which passes through the equinoxes. The solstitial colure is the circle of decli- nation which passes through the solstices. The equinoxes are the poles of the solstitial colure. By the annual motion of the earth, its axis is carried very nearly parallel to itself, so that the plane of the equator, which is always at right angles to the axis, is very nearly a fixed plane of the celestial sphere. The axis is, however, subject to small changes of direction, the effect of which is to change the position of the intersection of the equator and the ecliptic, and hence, also, the position of the equinoxes. Li expressing the positions of stars, referred to the vernal equinox, at any given instant, the actual position of the equinox at the instant is understood, unless otherwise stated. The right ascension of a point of the sphere is the arc of the equator intercepted between its circle of declination and the vernal equinox, and is reckoned from the venial equinox east- ward from 0° to 360°, or, in time, from 0* to 24\ The point of observation being supposed at the centre of the earth, neither the declination nor the right ascension will be affected by the diurnal motion: so that these co-ordinates are wholly independent of the obsers'^er^s position on the surface of the earth. Their values, therefore, vary only with the time, and are given in the ephemeridcs as functions of the time reckoned at some assumed meridian. We shall generally denote right ascension by a. As its value reckoned towards the east is positive, a negative value resulting 24 THE CELESTIAL SPHERE. from any calculation would be interpreted as signifying an arc of the equator reckoned from the vernal equinox towards the west. Thus, a point whose right ascension is 300®, or 20*, may also be regarded as in right ascension — 60®, or — 4* ; but such negative values are generally avoided by adding 360®, or 24*. Again, in continuing to reckon eastward we may arrive at values of the right ascension greater than 24*, or greater than 48*, etc.; but in such cases we have only to reject 24*, 48*, etc. to obtain values which exj^ress the same thing. 6. Fourth system of co-ordinates, — Celestial latitude and langi- tude, — In this system the ecliptic is taken as the primitive circle, and the secondaries by which points of the sphere are referred to it are called circles of hititude. Parallels of latitude are small circles parallel to the ecliptic. The latitude of a point of the sphere is its distance from the ecliptic mea8ured on a circle of latitude, and its longitude is the arc of the ecliptic intercepted between this circle of latitude and the vernal equinox. The longitude is reckoned eastward from 0® to 3G0®. The longitude is sometimes expressed in si(/ns^ degrees, &e., a sign being equal to 30®, or one-twelfth of the eclii)tic. These co-ordinates are also independent of the diurnal motion. It is evident, however, that the system of declination and right ascension will be generally more convenient, since it is more directly related to our first and second systems, which involve the observer's position. AVe shall denote celestial latitude by /9; longitude by L Posi- tive values of ^3 belong to j)oints on the same side of the ecliptic as the north [jole; negative values, to those on the oi»posite side. In connection with this system we may here define the nana- gcsiintd, which is that point of the ecliptic which is at the greatest altitude above the horizon at any given time. That vertical circle of the observer which is jierpendicular to the ecliptic meets it in the nonagesimal : and, being a secondary to the ecliptic, it is also a circle of latitude: it is the great circle which passes through the observer's zenith and the pole of the ecliptic. 7. Co-ordinates of the ohsnTers position. — We have next to ex- press the position of the obser\er on the surface of the earth, according to the different systems of co-ordinates. This will bo SPHERICAL CO-ORDIXATES. 25 done by referring his zenith to the primitive circle in the same manner as in the case of any other point. In the first system, the primitive circle being the horizon, of which the zenith is the pole, the altitude of the zenith is always 90°, and its azimuth is indeterminate. In the second system, the declination of the zenith is the same as the terrestrial latitude of the observer, and its hour angle is zero. The declination of the zenith of a place is called the geographical latitude^ or simply the latitude, and will be hereafter denoted by f. North latitudes will have the sign + ; south latitudes, the sign — . In the third system, the declination of the zenith is, as before, the latitude of the observer, and its right ascension is the same as the hour angle of the vernal equinox. In the fourth system, the celestial latitude of the zenith is the same as the zenith distance of the nonagesimal, and its celestial longitude is the longitude of the nonagesimal. It is evident, from the definitions which have been given, that the problem of determining the latitude of a place by astro- nomical observation is the same as that of determining the declination of the zenith ; and the problem of finding the lon- gitude may be resolved into that of determining the right ascension of the meridian at a time when that of the prime meridian is also given, since the longitude is the arc of the equator intercepted between the two meridians, and is, conse- quently, the diflerence of their right ascensions. 8. The preceding definitions are exemplified in the following figures. Fig. 1 is a stereographic projection of the sphere upon the plane of the horizon, the projecting point being the nadir. Since the planes of the equator and horizon are both perpendicular to that of the meridian, w\^ their intersection is also peri^endicular to it; and hence the equator WQE passes through the east and west points of the horizon. All vertical circles passing through the projecting point will be projected into straight lines, as the meridian NZS, the prime vertical WZE, and the vertical circle ZOH drawn through any point of the surface 26 THE CELESTIAL SPUEBE. of the sphere. We have then, according to the notation adopted in the first system of co-ordinates, h ^ the altitude of the point ^= OS, C ^ the zenith distanco " ^^ OZ, A = the azimuth " = SM, or = the angle SZH. If tlie declination circle POD be drawn, we have, in the second aystcin of co-ordinates, 3 ^ the declination of ^= OD, P = the polar distance " = PQ, I = the hour angle " = ZP2), or = QD. If Vis the vernal equinox, wo liavo, iii tlie third system of co-ordinates, S = the declination of = OB, a = the riglit aeconBion = VD, or = the angle VPD. In this figure is also drawn the six hour circle EPW, or the declination circle passing through the east and west points of the horizon. The angle ZPW, or the arc QW, heing 90°, the hour angle of a point on this circle is either + 6* or —6*, that is, either 6* or 18*. Fig. 2 is a repetition of the preceding figure, with the addi- ng. :. tion of the ecliptic and the circles related to it. C Vr re]) resents the ecliptic, P' its pole, P'OX a circle of latitude. Ilonce we have, in our third system of co-oi-dinates. ^ = the celestial latitude of — OL, i ^^ " longitude " :^ VL, ^ the angle VP'L. ■\Ve have also FPthc eqninoetial colure,P'7'.4iJ the solstitial coliirc. P'ZGF the vcrticiil circle passing through i", which is therefore peri>ondicnlar to the ecliptic at G. The point G is the nonagesimal ; ZG is its zenith distance, Vtf its longitude; or ZG is the celestial latitude and VG the celestial longitude of the zenith. Finally, we have, in hoth Fig. 1 and Fig. 2, fi = the geograpliicnl latitude of the observer = ZQ -- 00" - PZ^ PX SPHERICAL CO-ORDINATES. 27 Hence the latitude of the observer is always equal to the alti- tude of the north pole. For an observer in south latitude, the north pole is below the horizon, and its altitude is a negative quantity: so that the definition of latitude as the altitude of the north pole is perfectly general if we give south latitudes the negative sign. The south latitude of an observer considered independently of its sign is equal to the altitude of the south pole above his horizon, the elevation of one pole being always equal to the depression of the other. 9. Numerical expression of hour angles. — The equator, upon which hour angles are measured, may be conceived to be divided into 24 equal parts, each of which is the measure of one hour, and is equivalent to ^ of 360°, or to 15°. The hour is divided sexagesimally into minutes and seconds of time^ distinguished from minutes and seconds of arc by the letters "* and ' instead of the accents ' and ". We shall have, then, 1* = 15° 1- = 15' 1- = 15" To convert an angle expressed in time into its equivalent in arc, multiply by 15 and change the denominations * "* * into ° ' ''; and to convert arc into time, divide by 15 and change ° ' " into * »» •. The expert computer will readily find ways to abridge these operations in practice. It is well to observe, for this purpose, that from the above equalities we also have, lo = 4- 1' = 4- and that we may therefore convert degrees and minutes of arc into time by multiplying by 4 and changing ° ' into "^ * ; and reciprocally. TRANSFORMATION OF SPHERICAL CO-ORDTNATES. 10. Given the altitude (h) and azimuth {A) of a star, or of any point of the sphere, and the latitude (^) of the observer, to find the declina- tion (3) and hour angle {t) of the star or the point. In other words, to transform the co-ordinates of the first system into those of the second. This problem is solved by a direct application of the formulae of Spherical Trigonometry to the triangle POZ, Fig. 1, in which, being the given star or point, we have three parts given, 28 THE CELESTIAL SPHERE. Fig. 3. namely, ZO the zenith distance or complement of the given altitude, PZO the supplement of the given azimuth, and PZ the complement of the given latitude ; from which we can find PO the jv)lar distance or complement of the required declination, and ZPO the required hour angle. But, to avoid the trouble of taking complements and supplements, the fomiulfe are adapted 8o as to ex[)ress the deelinati<m and hour angle directly in terms of the altitude, azimuth, and latitude. To show as clearly as possible how the formulje of Spherical Trisconometrv are thus converted into fonnuhe of Spherical Astronomy, let us first con- sider a spherical triangle ABC, Fig. 3, in which there are given the angle A, and the sides 6 and e, to find the angle B and the side a. The general rela- tions between these five quantities are [Sph. Trig. Art. 114]* cos a = cos c cos h -\- sin c sin b cos A ^ sin a cos B r^ sin c cos b — cos c sin b cos A > ((X) sin a sin B -~ sin b sin A J Now, comparing the triangle ABC with the triangle PZO of Fig. 1, we have A r^ PZO r= 180° — .4 b=^ ZO^ 90°— h c= PZ= 90°— sp (3=^ PO:^90° — ^ B ^ PZO --^ t Substituting these values in the above equations, we obtain sin ^ - sin 9" sin h — cos ^ cos h cos A (1) cos d cos t r^ cos if H\n h -{- sin ^ cos /* cos A (2) cos d sin t ~- cos h sin A (3) which are the required expressions of o and t in terms of h and A, If the zenith distance (^) of the star is given, the equations will be sin o - : sin <p cos Z — cos if sin C cos A (4) cos a cos t —-. cos <f cos C + sin ^ sin r cos A (5) cos «J sin f 1 - sin C sin ^l (<>) Since, in Spherical Astronomy, we consider arcs and angles whose values mav exceed 180^, it becomes necessary, in general. ♦ The references to Trigonometry are to the oth edition of the author's *'Trcnlisc on Plane and Spherical Trigonometry/' SPHERICAL CO-ORDINATES. 29 to determine such arcs and angles by both the sine and the cosine, in order to fix the quadrant in which their vahies are to be taken. It has been shown in Spherical Trigonometry that when we consider the general triangle, or that in which values are admitted greater than 180°, there are two solutions of the triangle in every case, but that the ambiguity is removed and one of these solutions excluded " when, in addition to the other data, the sign of the sine or cosine of one of the required parts is given." [Sph. Trig. Art. 113.] In our present problem the sign of cos 8 is given, since it is necessarily positive ; for d is always numerically less than 90°, that is, between the limits +90° and —90°. Hence eos t has the sign of the second member of (2) or (5), and sin t the sign of the second member of (3) or (6), and t is to be taken in the quadrant required by these two signs. Since h also falls between the limits +90° and —90°, or ^ between 0° and 180°, cos A, or sin ^, is positive, and therefore by (3) or (6) sin t has the sign of sin A ; that is, when A < 180° we have t < 180°, and when A>180° we have ^ > 180°,— conditions which also follow directly from the nature of our problem, since the star is west or east of the meridian according as A < 180° or A > 180°. The formula (1) or (4) fully determines 5, which will always be taken less than 90°, positive or negative according to the sign of its sine.* To adapt the equations (4), (5), and (6) for logarithmic compu- tation, let m and M be assumed to satisfy the conditions [PL Trig. Art. 174], m sin Mz= sin C cos A 1 . m cos Jf = eos C / the three equations may then be written as follows : sin ^ = wi sin (<p — M) eos d COS t = m cos (^ — ilf) \ (8) cos ^ sin ^ = sin C sin A K we eliminate m from these equations, the solution takes the following convenient form : * There are, however, special problems in which it is convenient to depart from this general method, and to admit declinations greater than 90°, as will be seen hereafter. 30 THE CELESTIAL SPHERE. tan Jlf = tan C cos A tan A sin M tan t = \ (^\ cos (jp — J[f ) ' ^^^ tan ^ = tan (tp — M) cos t in the use of which, we must observe to take t greater or less than 180° according as A is greater or less than 180°, since the hour angle and the azimuth must fall on the same side of the meridian. Example. — ^In the latitude f = 38° 58' 53", there are given for a certain star C =- 69° 42' 30", A = 300° 10' 30" ; required d and L The computation by (9) may be arranged as follows :* log tan C 0.4320966 ^ r^ 38® 68^ bS'^ log COB A 9.7012595 log tan A ii0.2866026 M= 53 39 41.98 log tan i/^ 0.133356]_ log sin .If 9.9060828 ^— Jf= — 14 40 48.98 logtan (0— if) n9!4182633 log sec (^—M) 0.0144141 /= 304 55 26.49 log cos ^ 9.7577677 log tan t fiO.1559995 i5= —8 81 46.50 log tan (J n9.1760310 Converting the hour angle into time, we have ^ = 20* 19* 4r.766. 11. The angle POZy Fig. 1, between the vertical circle and the declination circle of a star, is frequently called the jxiraUactic angle y and will here be denoted by q. To find its value from the data ^, -4, and ^, we have the equations cos d cos ^ = sin C sin ^ + oos C cos f cos A \ ^i a cos d sin q = cos ip sin A ) which may be solved in the following form : / sin F=:zf<\n ^ f cos F=^ cos C cos A cos d cos q =f cos (^ — F) cos d sin ^ = cos ^ sin A or in the following : g sin (r = sin 9 g cos G -- cos ^ cos A cos d cos q = g cos (C — G) cos d sin q = cos ^ sin A or again in the following : (11) (12) * In this work the letter n prefixed to a logarithm indicates that the number to which it corresponds is to hare the negatiye sign. SPHERICAL CO-ORDINATES. 81 tan G = tan f sec A tan jr = tan A cos G v (13) COS (C — G) and, in the use of the last form, it is to be observed that q is to be taken greater or less than 180° according as ^ is greater or less than 180°, as is evident from the preceding forms. 12. If, in a given latitude, the azimuth of a star of known declination is given, its hour angle and zenith distance may be found as follows. We have cos t sin ^ — sin t cot A = cos <p tan d cos C sin ^ — sin C cos tp cos A=^mi d The solution of the first of these is effected by the equations & sin J? = sin ^ h cos B = cot A sm (jB — ^) = ^ and that of the second by c sin C= sin <p c cos C= cos fp cos A sm (C — C) = c 13. Finally, if from the given altitude and azimuth we wish to find the declination, hour angle, and parallactic angle at the same time, it will be convenient to use Gauss's Equations, which for the triangle ABC, Fig. 3, are cos } a sin } (B + C) = cos } (6 — c) cos } A cos } a cos } (B + C ) = cos } (6 + c) sin J A . .^ sin } a sin } (B — C) = 8in } (6 — c)cos}A ' ^ ^ sin J a cos } (B — C) = sin } (6 + c) sin } A which are to be solved in the usual manner [Sph. Trig. Art. 116] after substituting the values A = 180° — -4, 6 = C> <? = 90° — ^, a = 90° — 5, B = /, C = q. 14. Given the declination {8) and hour angle (t) of a staVy and the latitude (^), to find the zenith distance (^) and azimuth {A) of the star. That is, to transform the co-ordinates of the second system into those of the first. We take the same general equations (iE) of Spherical Trigo- nometry which have been employed in the solution of the pre- 32 THE CELESTIAL SPHERE. ceding problem, Art. 10; but we now suppose the letters A, B, C, in Fig. 3, to represent respectively the pole, the zenith, and the star, so that we substitute b = W — d B = 180<' — A c = 90^ — ^ and the equations become cos C = sin f sin d -\- cos f cos d cos t sin C cos A=z — cos ^ sin ^ -f- sin ^ cos d cos t y (14) sin C sin A = cos ^ sin t which express ^ and A directly in terms of the data. Adapting these for logarithmic computation, we have m sin M= sin d m cos M= cos 3 cos t cos Z =m cos (^ — M) ) (15) sin Z cos A = m sin (^ — M) sin C sin A = cos d sin t in which m is a positive number. Eliminating m, we deduce the following simple and accurate formulas : ,-. tan^ tanif = cos t . tan t cos if . .^^. tan A = -7— 5^. ) (16) sin (vp — M) tan C = ^'^"^^-^) COS A where A is to be taken greater or less than 180° according as t is greater or less than 180°. Example 1.— In latitude ^ = 38° 58' 53", there are given for a certain star, 5-= -8° 31' 46".5G, / = 20* 19- 41'.766; required A and ^. By (16) we have : log tan 6 fi0.l7A0310 ^ = 88« 68^ 53'^ log cos t 9. 7677077 log tan t wO. 1659995 ir=— 14 40 48.98 log tan .V fi0^182«88 log cos if 9.9865859 p^M ^ 5389^1.98 log tan (^— -V) 0.1333561 log co««c (^ — 1^)^0989172 A=z 800 10 80 log cos A 9.70_12r)95 log tan A fiO.2356026 C iss C9 42 80 log tan C 0.4320906 SPHBRICAL CO-ORDINATES. 83 For verification we can use the equation sin C sin A =: cos d sin t log sine 9.9721748 log cos ^ 9.9951697 log sin A 9.9367621 log sin t 9.9137672 9.9089369 9.9089369 Example 2. — In latitude y>= —48^ 32', there are given for a star, 5 = 44° 6' 0", i = 17* 25"» 4* ; required A and C- We find A = 241° 53' 33".2, C = 126° 25' 6".6 ; the star is below the horizon, and its negative altitude, or depression, is A = — 86° 25' 6".6. K the zenith distance of the same star is to be frequently com- puted on the same night at a given place, it will be most readily done by the following method. In the first equation of (14) substitute cos t = l — 2 sin' } t then we have cos C = cos {9-rd) — 2 cos f cos d sin" } t where ip>rd signifies either ip — d ord — <pj and if ^ > ^ the latter form is to be used. Subtracting both members from unity, we obtain sin' } C = sin' J {9-^^) + cos f cos d sin' } t Now let m = i/cos ip cos d n = 8in } (sp*^^) then we have 8inK = nJT7^5?^^ ^ n' and hence, by taking an auxiliary N such that tan N= — sin J ^ n we have ) (17) sin K = Ty=- — ^sin it cos iV Bin N The second form for sin \ f will be more precise than the first when sin iVis greater than cos N. The quantities m and n will be constant so long as the decli- nation does not vary. 16. If the parallactic angle q (Art. 11) and the zenith distance Voi-L— 3 u THE CELESTIAL SPHERE. j; an- rc-quired from the data y>, *, and <, they may be found fr^jiu the equations cos C = sin f sin d -f- cos f cos ^ cos t sio C cos ^f = sin ^ cos ^ — cos tp sin ^ cos < V (18) sin C sin ^ = cos ^ sin t which are adapted for logarithms as follows : n sin iV= cos ^ cos ^ ncosiV=8in ^ cos C = n sin (^ + iV) ^ (19) sin C cos q=zn cos \p + JV) sin C sin ^ = cos s^.sin t Off eliminating n, thus: tan JV=cot ^ cost . . tan ^ sin iV , tan C sin flr = \ (-20^ ^ sin {d +N) ' ^ ^ tan C COH q =: cot (^ + iV^ VVIjirn this last fonu is employed in the case of a star which UiiH \n:i'U olmerved above the horizon, tan (^ is known to be posi- tive, and there is no ambiguity in the determination of q. This inrtn iH, tlierefore, tbe moHt convenient in practice. If f. A, and q are all rcciuired from the data 5, t, and ^, we luive, by (iauHn's eijuationH, Hin } C Hin J (A + q)=^ sin } ^ cos } (^ + d) sin i C COM i (A + q) -- com § ^ sin J (^ — d) iOH i C Hin } (/t — q) ~ sin it sin i (f + ^ <!OH i C eo8 1 (A — /jr) — cos M cos } (^p — d) (21) fiK. 4. 10. Wlii'n tbi! altitude, azimuth, and parallactic angle of known stars are to be frequently computed at the same place, the labor of computation is much diminished by an auxiliary table pre- jmred for the latitude of the place accord* ing to formulic proposed by Gauss. A Hpecimen of such a table computed for the latitude of the Altona Observatorv will be found in ^^&humacher's IlUlfstafcbiy neu lieniusg. V. AV'^arnstorif/* The requisite formnhe are readily deduced as follows: Let tli4» d<»< lination circle through the object 0, Fig. 4, be produced to intersect the horizon in F. By the diurnal motion SPHERICAL CO-ORDINATES. 35 the point F changes its position on the horizon with the time ; hut its position depends only on the time or the hour angle ZPO^ and not upon the declination of 0, The elements of the position of F may therefore be previously computed for succes- sive values of L We have in the triangle PFSj right-angled at Sj FPS==t, PS= 180° — ip; and if we put 1Si = FSj J5 = P2?^— 90% y = 180° -P2?:a we find tan a = sin 9» tan t, tan B = cot f cos t, cot ^ = Bm B tan t We have now in the triangle HOF^ right-angled at H, B + d=OF, r = HFO, h = OS, and if we put u = SF = SS''FS = A--% we find tan u = cos T' tan (^B + d) A = ^ + u sin A = sin T' sin {B -f d) or, tan h = tan y sin u. To find the parallactic angle q=POZj we have in the triangle HOF tan q = cot y sec (J? + d) In the Gaussian table for Altona as given in the "Hiilfstafeln" we find five columns, which give for the argument t, the quan- tities a. By log cos Yy log sin y, log cot y, the last three under the names log (7, log D, and log -B", respectively. With the aid of this table, then, the labor of finding any one of the quan- tities hy Ay q IB reduced to the addition of two logarithms, namely: tan u= Ct&n (^B + d) sin h = D sin (B + d) A = a + t« tangr = j&8ec (B+d) The formulae for the inverse problem (of Art. 10) may also be found thus. Let G be the intersection of the equator and the vertical circle through 0, and put B = HGy u=DGy^=QGy Y = ZGQ; then we readily find tan 9i = sin f> tan A, tan B = cot ^ cos A, cot Y = QmB tan A which are of the same form as those given above, with the ex- change of A for L Hence the same table gives also the elements of the point Gy by entering with the argument "azimuth,** ex- pressed in time, instead of the hour angle. We then have ^ = 36 THE CELESTIAL SPHERE. DQ, and if we here put u = DG = ^ — t, we have from the triangle GDO sin J = sin ;* sin (h — B) tan u = cos y^ tan (h — B) or, employing the notation of the table, tan M = (7 tan (A — B) sin ^ = D sin (h — B) t = ^ — u tan ^ = J& see (A — B) 17. To find the zenith distance and azimuth of a star^ when on the six hour circle. — Since in this case < = 6* = 90°, the triangle PZOy Fig. 4, is right-angled at P, and gives immediately cos ZO = cos PZ cos PO cot PZO = sin PZ cot PO or, since PZO = 180° — J, and cot PZO = - cot A, cos C = sin ^ sin d cot A= — cos ^ tan d But if the star is on the six hour circle east of the meridian, we must put t = 18*= 270° and PZ 0=^ — 180° ; hence for this case cot A = -f cos ^ tan d A more general solution, however, is obtained from the equa- tions (14), by putting cos < = 0, sin < = ± 1, whence cos C = sin 9> sin d '\ sin C cos A = — cos ^ sin J V (22) sin C sin A = rb cos d ) the lower sign in the last equation being used when the star is east of the meridian. Example. — Required the zenith distance and azimuth of Sirius, 8 = — 16° 31' 20", when on the six hour circle east of the meri- dian at the Cape of Good Hope, ^ = — 33° 56' 3". We find log (—cos 8) = log sin : sin A =n9.9816870 log (— cos ^ sin d) = log sin C cos il = 9.3728204 A = 283°49'84".9 log sin A = 9.9872302 log sin C = 9.9944568 log sin 9> sin a = log cos C = 9 .2007309 C =~80° 5r 55" SPHERICAL CO-ORDINATES. 87 Fig. 6. 18. To find the hour avgle^ azimuihy and zenith distance of a given star at its greatest elongation. — In this case the vertical circle ZSj Fig. 5, is tangent to the diurnal circle, SAj of the star, and is, therefore, perpendicular to the declination circle PS. The right triangle PZS gives, therefore, tan ^ cos t = BmA = C08C = tan d cos 9 cos ^ sin ^ sin S (23) If d and <p are nearly equal, each of the quantities cos t, sin Ay and cos ^ will be nearly equal to unity, and a more accurate solution for that case will then be as follows : Subtract the square of each from unity ; then we have tan' d — tan* ^ sin (d -f ^) sin (d — ^) sin* t = eoB"il = Bin' C = tau'^ cos' — cos* d 7* cos'^ sin'^ — sin'^ cos'^ sin'^ sin (d 4- 9) 81^ (^ — 9) cos'^ sin (d + 9) sin (d — ^) sin' d sin' d Hence if we put k = y/[8in (d + 9») sin (d — ^)] we shall have k . k Bmt = COB A = sin C = sin 3 (24) cos ^ sin d cos ^ 19. To find the hour angky zenith distance^ and parallactic angle of a given star on the prime vertical of a given place. In this case, the point in Fig. 1 being in the circle WZE^ the angle PZO is 90®, and the right triangle PZO gives tan d cos t = cosC = smgr = tan ^ sin d sin ^ cos 9> cos d (25) 38 THE CELESTIAL SPHERE. If 5 is but little less than y>, the star will be near the zenith, and, as in the preceding article, we shall obtain a more accurate solution as follows : Put k = i/[8in (f + d) sin {<p — dj] then Bin C = —, cos q = ^^^^ (26) sin t = sin ip cos d %in<p cos^ We may also deduce the following convenient and accurate formula for the case where the star's declination is nearly equal to the latitude [see Sph. Trig. Arts. 60, 61, 62] : tan ht =z tan } C = // sin (y — d) \ Vein (sp + d) I /tan J (sp — ^) ) tan } (f> + i)/ tan (45" — i q) = ^[tan i (f + S) tan } (^ (27) -»)] If 5 > ^, these values become imaginary; that is, the star can- not cross the prime vertical. Example. — Required the hour angle and zenith distance of the star 12 Canum Venaticorum [d = -\- 39°»5' 20") when on the prime vertical of Cincinnati (^ = + 39° 5' 54"). ^ — a = 0° 0' 34" ^ + ^ = 78 11 14 log sin {<p — d) 6.21705 log sin lip + ^)_9.99070^ 2)6722635 log tan } t 8.11318 i f = 0° 44' 36".6 t = 1° 29' 13".2 = 0* 5- 56*.88 }(^ — ^)= 0^0' 17" i (ip -^d)z= 39 5 37 log tan i (^ —d) 5.91602 log tan 1 (^ + ^) 9.90982 2)6^00620 log tan i C 8.00310 i C = 0° 34' 37".3 : = r 9'14".6 20. To find the amplitude and hour angle of a given star ichen in the horizon. — If the star is at H, Fig. 1, we have in the triangle PILY, right-angled at N, PN ^ 9. HPN ^. 180° - U PH ^= 90° — d\ and if the amplitude WH is denoted by a, we have HN= 90° — a. This triangle gives, therefore, sin a =1^ 8oc ^ sin d cos f =: — tan tp tan d } (28) SPHEBICAL CO-ORDINATES. 89 21. Given the hour angle {t) of a siar^ to find its right ascension (a). — Transformation from our second system of co-ordinates to the third. There must evidently be given also the position of the meridian with reference to the origin of right ascensions. Suppose then in Fig. 1 we know the right ascension of the meridian, or VQ = 0, then we have VD = VQ —DQ, that is, a= e —t Conversely, if a and are Igiown, we have f = e — a The methods of finding at a given time will be considered hereafter. 22. Given the zenith distance of a knovm star at a given place, to find the star's hour angle^ azimuth^ and parallactic angle. In this case there are given in the triangle POZj Fig. 1, the three sides ZO = C, POP = 90° -8, PZ= 90° - ^, to find the angles ZPO = t, PZO = 180° — A, and POZ =q. The formula for computing an angle B of a spherical triangle ABC, whose sides are a, 6, c, is either sin J B = J / 8in(^-^)Bin(^-c) \ \ \ sin a sin <? / C08iB = ^/f «'"""'" (^-^) \ Af \ sin a sin c / or f^n^T^-- // 8in(5-a)sin(^-c) \ \ \ sin 5 sin (s — b) I in which 5 = J (a + 6 + <?)• We have then only to suppose B to represent one of the angles of our astronomical triangle, and to substitute the above corresponding values of the sides, to ob- tain the required solution. This substitution will be carried out hereafter in those cases where the problem is practically applied. 23. Given the declination (8) and the right ascension (a) of a star, and the obliquity of the ecliptic (e), to find the latitude (fi) and the longi- tiule {X) of the star. — Transformation from the third system of co- ordinates to the fourth. The solution of this problem is similar to that of Art. 10. 40 THE CELESTIAL SPHERE. Fig. 6. The analogy of the two will be more apparent if we here repre- sent the sphere projected on the plane of the equator as in Fig. 6, where VBUCia the equator, P its pole ; VA U the ecliptic, P' its pole, and consequently CP'PB the solstitial colure; POD^ P'OL, circles of 1^ declination and latitude drawn through the star 0. Since the angle which two great circles make with each other is equal to the angular distance of their poles, we have PP' = £ ; and since the angle P'PO is measured by CD and PP'Ohy AL^ we have in the triangle PP'O P'PO, PP'O, P'O, PO, PP' 90^ + a, W" — X, 90<> — Py 90*» — ^, c which, substituted respectively for A, B a, *, in the general equations (31), Art. 10, give sin /9 = cos e sin ^ — sin e cos d sin a cos ^ sin >l = sin e sin ^ -f- cos e cos d sin a cos P cos X = cos d cos a (29) which are the required formulae of transformation. Adapting for loe^rithmic computation, we have m sin J[f = m cos M = sin /9 = cos fi B\n X =z cos fi cos >l = sin d cos d sin a m sin (M- m cos (M - cos S cos a (80) in which m is a positive number. A still more convenient form is obtained by substituting k = m cos d k' = cos ^ m by which we find SPHERICAL CO-ORDINATES. 41 k sin M k cos JIf X/sin >l A:' cos >l tan fi cos fi sin X tan d sin a cos ( Jf — e) cos Jlif cot a sin X tan (Jlf cos (M — e) (31) For verification : coB^Bina = coaJtf Example. — Given ^, a, and e as below, to find ^ and X. Com- putation by (31). a = — 16° 22' 35".45 a = 6 33 29 .30 e= 23 27 31.72 log tan ^ = log k sin M n9.4681562 log sin a = log A: cos M 9.0577093 jf 3= _ 68*» 45' 41".87 JM"— e = — 92 13 13 .59 log cos M 9.5590070 log cot a 0.9394396 log A' cos >l 0.4984466 logcos (-Sf — e) = log A-' sin >l n8.5882080 X = 359*> 17' 43".91 log sin >l W8.0897286 log tan (ilf— e ) 1.4114658 log tan ^ n9.5011944 /5 = — 17° 35' 37".51 Verificaiion. log cos fi sin X n8.0689234 log cos d sin a 9.0397224 log o_08(Jtf-e) „9 02g20i(^ COS Jf Tables for facilitating the above transformation, based upon the same method as that employed in Art. 16, are given in the American Ephemeris and Berlin Jahrbuch. The formulfe there used may be obtained from Fig. 6, in which the points F and G are used precisely as in Fig. 4 of Art. 16. 24. K we denote the angle at the star, or P'OP, by 90° — Ey the solution of the preceding problem by Gauss's Equations is cos (45° cos (45° Bin (45° sin (45° L5)8inJ(J5;+>l) i,3)(^osi(E+xy i/9)sin HE—Xy ifi)cosilE—Xy : sin [45° cos [45° sin [45° cos [45° }(e_^)]8in(45°+}a) i (e + ^)] cos (45° + } a) } (e + d)] COS (45° + J a) J(e — ^)]sin(45° + Ja) (32) 25. If the angle at the star is required when the Gaussian Equations have not been employed, we have from the triangle jPOP', Fig. 6, putting P'OP = 7i = W — JS, 42 THE CELESTIAL SPHERE. COS /5 COS Tj = COS e COS ^ -|- sin € sin d sin a COS ^ sin 17 ^ sin e cos a or, adapted for logarithms, n sin iV=ir sin e sin a n cos iV = cos e cos ^ cos ly = n cos (N—d) ( (^) cos fi Bin Tj =z sin e cos a 26. Given the latitude (j9) and longitude {X) of a star^ and the obliquity of the, ecliptic (e), to find the declination and right ascension of the star. By the process already employed, we derive from the triangle FP'O, Fig. 6, for this case, sin d = cos e sin ^ + sin e cos fi sin X cos ^ sin a = — sin € sin ^ -|- cos e cos ^ sin A )- (34) cos 6 cos a = cos fi cos >l which, it will be obsen-ed, may be obtained from (29) by inter- changing a with Xy and 8 with ^, and at the same time changing the sign of e, that is, putting — e for e, and, consequently, — sin e for sin e. For logarithmic computation, we have m sin JW= sin ^ m cos M = cos ^ sin X sin d = m sin (3/ + c) ) (36) cos ^ sin a = m cos {M + e) cos d cos a = COS /5 COS X or the following, analogous to (31) : A* sin M = tan ff k cos ^f= sin >l k' sin a = cos (M + c) A'' cos a =r-. CO8 Jf cot A tan d -- sin a tan (-If + t) __ . , . coH r) sin a cos (^f -f c") For verification : k: •— t= — . " ir — -^ cos /Sf sm >l cos M (36) 27. The angle at the star, POP', being denoted, as in Art. 24, RECTANGULAR CO-ORDINATES. 48 by 90° — Ey the solutiou of this problem by the Gaussian Equations is sin (45<^— J d) sin \(JE+d) = sin [45°— i (e + ^)] sin (45°+ J X) sin (45°— J a) cos } (J5: + a) = cos [45°— } (e — ^5)] cos (45°+ } X) cos (45°— i d) sin } (jE — a) = cos [45°— } (e — /9)] sin (45°+ i A) cos (45°— } d) cos i (^ — a) = sin [45°— } (e + /9)] cos (45°+ } X) (37) 28. But if the angle yj = 90° — j& is required when the Gaussian Equations have not been employed, we have directly cos d cos Ti = cos € cos fi — sin s sin fi sin X cos ^ sin 17 = sin e cos X or, adapted for logarithms, n sin JV= sin c sin >l n cos iV= cos c cos ^ cos ly = n cos (iV + ^) [ (^^) cos ^ sin ly == sin € cos X 29. i^or <A€ sun, we may, except when extreme precision is desired, put /3 = 0, and the preceding formulae then assume very simple forms. Thus, if in (34) we put sin /3 = 0, cos /? = 1, we find sin ^ = sin e sin X cos ^ sin a = cos e sin X cos d cos a = COS X whence if any two of the four quantities 5, a, ^, e are given, we can deduce the other two. RECTANGULAR CO-ORDINATES. 30. By means of spherical co-ordinates we have expressed only a star's direction. To define its position in space com- pletely, another element is necessary, namely, its distance. In Spherical Astronomy we consider this element of distance only so far as may be necessary in determining the changes of apparent direction of a star resulting from a change in the point from which it is viewed. For this purpose the rectangular co- ordinates of analytical geometry may be employed. Three planes of reference are taken at right angles to each other, their common intersection, or origin, being the point of 44 THE CELESTIAL SPHERE. obserration; and the star's distances from these planes are denoted by x^ y, and z respectively. These co-ordinates are respectively parallel to the three axes (or mutual intersections of the planes, taken tvvo and two), and hence these axes are called, respectively, the axis of x, the axis of y, and the axis of z. The planes are distinguished by the axes they contain, as "the plane of xy,'' the "plane of xz," the "plane of y^/' The co- ordinates may be conceived to be measured on the axes to which they belong, from the origin, in two opposite directions, distinguished by the algebraic signs of Tplus and minus^ so that the numerical values of the co-ordinates of a star, together with their algebraic signs, fully determine the position of the star in space without ambiguity. Of the eight solid angles formed by the planes of reference, that in which a star is placed will always be known by the signs of the three co-ordinates, and in one only of these angles will the three signs all be plus. This angle is usually called the first angle. To simplify the investigations of a problem, we may, if we choose, assume all the points considered to lie in the first angle, and then treat the equations obtained for this simplest case as entirely general; for, by the principles of analytical geometry, negative values of the co-ordinates which satisfy such equations also satisfy a geometrical construction in which these co-ordinates are drawn in the negative direction. The polxir co-ordinates of analytical geometry (of three dimen- sions) when applied to astronomy are nothing more than the Hphcrical co-ordinates we have already treated of, combined with the element distance ; and the formuhe of transformation from rectangular to polar co-ordinates are nothing more than the values of the rectangular co-ordinates in terms of the dis- tance and the spherical co-ordinates. For the convenience of reference, we whall here recapitulate these fornmla?, with special reference to our several systems of spherical co-ordinates. 31. We shall find it useful to premise the following Lemma. — The distanre of a point in space from the plane of ant/ great circle of the celestial sphere is equal to its distance from the centre of the sphere multiplied by the cosine of its angular distance from the pole of that circle; and its distance from the axis of the circle is equal to its distanre from the centre of the sphere multiplied by the sine of its angular distance from the pole. RECTANGULAR CO-ORDINATES. 45 For, let ABy Fig. 7, be the given great circle orthographi- cally projected upon a plane passing through its axis OP and the given point C; P its pole. The dis- tance of the point C from the plane of the great circle is the perpendicular CD ; CE is its distance from the axis; CO its dis- tance from the centre of the sphere; and the angle COP the angular distance from the pole. The truth of the Lemma is, therefore, obvious from the figure. B 82. The values of the rectangular co-ordinates in our several systems may be found as follows : First system. — Altitude and azimuth. — ^Let the primitive plane, or that of the horizon, be the plane of xy; that of the meridian, the plane of xz; that of the prime vertical, the plane of yz. The meridian line is then the axis of x; the east and west line, the axis of y; and the vertical line, the axis of z. Positive x will be reckoned from the origin towards the south, positive y towards the west, and positive z towards the zenith. The first angkj or angle of positive values, is therefore the southwest quarter of the hemisphere above the plane of the horizon. Let Z, Fig. 8, be the zenith, S the south point, W the Fig. 8. west point of the horizon. These points are respectively the poles of the three great circles of reference ; if, then, A is the position of a star on the surface of the sphere as seen from the centre of the earth, and if we put h = altitude of the star = AH, A = azimuth « = SHy J = its distance from the centre of the sphere we have immediately, by the preceding Lemma, X = J cos ASf y = J cos AWy z = A cos AZ^ which, by considering the right triangles AHSj AHWj become a: =r J cos h co8 A \ y = A cos A sin A > (39) j2r =: J sin i^ J These equations determine the rectangular co-ordinates x^y^z. 46 TUE CELESTIAL SPHERE. when the polar co-ordinates J, A, A are given. Conversely, if Xy y, and z are given, we may find J, A, and A; for the first two equations give tan A = — X and then we have J sin A = 2: J cos A = — ^ — = U. cos A sin A whence A and A. Or, by adding the squares of the first two equations, we have J cos h = y^x* + y' whence tan h = and the sum of the squares of the three equations gives J == y/^x" + y« + 2^) Second system. — Declination and hour angle. — ^Let the plane of the equator be the plane of xy; that of the meridian, the plane of xz; that of the six hour circle, the plane of i/z. In the pre- ceding figure, let Z now denote the north pole, S that point of the equator which is on the meridian above the horizon and from which hour angles are reckoned, TFthe west point. Posi- tive X \vi\\ be reckoned towards *§, positive y towards the west, positive z towards the north. If then A is the place of a star on the sphere as seen from the centre, and we put d =-. the star's declination = Aff, t = " hour angle = SH, A= " distance from the centre, and denote the rectangular co-ordinates in this case by x\ y', z'^ we have a/ rT= J cos d cos t y' = J cos ^ sin e y (40) y = J sin ^ Third system, — Declination and right ascension. — Let the plane of the equator be the phme of xy; that of the equinoctial colure, the plane of xz; that of the solstitial colure, the plane of yz. RECTANGULAR CO-ORDINATES. 47 The axis of x is the intersection of the planes of the equator and equinoctial colure, positive towards the vernal equinox ; the axis of y is the intersection of the planes of the equator and sol- stitial colure, positive towards that point whose right ascension is +90° ; and the axis of z is the axis of the equator, positive towards the north. K then, in Fig. 8, Z is the north pole, W the vernal equinox, A a star in the first angle, projected upon the celestial sphere, and we put H = doclination of the star = AH, a = right ascension " = WS, A = distance from the centre, while z'\ y"y z" denote the rectangular co-ordinates, we have af'= J cos ATF, y"= J cos AS, 2^' = A cos AZ, which become od' = A cos d cos a ^ y" = J cos ^ sin a > (41) 2^' = A sin d ) Fourth system. — Celestial latitude and longitude. — ^Let the plane of the ecliptic be the plane of xy ; the plane of the circle of latitude passing through the equinoctial points, the plane of xz ; the plane of the circle of latitude passing through the solstitial points, the plane of yz. The positive axis of x is here also the straight line from the centre towards the vernal equinox ; the positive axis of y is the straight line from the centre towards the north solstitial point, or that whose longitude is +90° ; and the positive axis of z is the straight line from the centre towards the north pole of the ecliptic. If then, in Fig. 8, Z now denotes the north pole of the ecliptic, W the vernal equinox, A the star's place on the sphere, and we put /9 = latitude of the star = AHy X = longitude of the star = WS, A = distance of the star from the centre, and x'", 2/'", z'", denote the rectangular co-ordinates for this system, we have a/" = A cos /5 cos X y"' = J cos /9 sin X } (42) y" = J sin /9 48 THE CELESTIAL SPHERE. TRANSFORMATION OF RECTANGULAR CO-ORDINATES. 33. For the purposes of Spherical Astronomy, only the most simple cases of the general transformations treated of in analy- tical geometry are necessary. We mostly consider but two cases: First. Transformation of rectangular co-ordinates to a new originj icithout changing the system of spherical co-ordinates. The general planes of reference which have been used in this chapter may be supposed to be drawn through any point in space without changing their directions, and therefore without changing the great circles of the infinite celestial sphere which repre- sent them. We thus repeat the same system of spherical co-ordi- nates with various origins and difterent systems of rectangular co-ordinates, the planes of reference, however, remaining always parallel to the planes of the primitive system. The transformation from one system of rectangular co-ordi- nates to a parallel system is evidently etfected by the formulae x^ = Xj, -f- a (43) in which rp y,, z^^ are the co-ordinates of a point in the primitive system ; r,, y,, z^ the co-ordinates of the same point in the new system ; and cr, 6, c are the co-ordinates of the new origin taken in the first system. As we have shown how to exj^ress the values of x^^ y^ z^ and of Tj, 2/2, ^2 "^ terms of the spherical co-ordinates, we have only to substitute these values in the preceding formula? to obtain the general relations between the spherical co-ordinates correspond- ing to the two origins. This is, indeed, the most general method of determining the eftect of parallax^ as will appear hereafter. Second. Transformation of rectangular co^ ordinates tchen the system of spherical co-ordi- nates is changed but the origin is unchanged. This amounts to changing the directions of the axes. The cases which occur in practice are chiefly those in which the two systems ^ have one plane in common. Suppose this plane is that of xz, and let OT, OZ, Fig. 9, be the axes of z and z iu the first system; 0-^^, Fig. y. BECTANGULAR CO-ORDINATES. 49 OZ^J the axes of Xj and z^ in the new system. Let A be the projection of a point in space upon the common plane; and let z = ABy 2 = OBy x^=AB^y z^=OBy The distance of the point from the common plane being unchanged, we have y=yv The axis of ^ may be regarded as an axis of revolution about which the planes of yx and yz revolve in passing from the first to the second system ; and if u denotes the angular measure of this revolution, or u = XOX^ = ZOZ^ = BAB^^ we readily derive from the figure the equation X sec u=^x^ — 2j tan u or, multiplying by cos u, x = Xi C08M — z^ sin u and 2 =x tan u-\-2i sec u or, substituting in this the preceding value of a:, 2 =zx^H\n u-^-z^ cos u Thus, to pass from the first to the second system, we have the formulae x = x^cosu — z^Binu '\ y = yi > (44) 2r = Xj sin w + z^ cos u ) And to pass from the second to the first, we obtain with the same ease. X^= X cos U-^-ZBlUU yt= y }- (45) 2^ = — iC sin M + zcos u As an example, let us apply these to transforming from our second system of spherical qo-ordinates to the first ; that is, from declination and hour angle to altitude and azimuth. The meri- dian is the common plane ; the axis of z in the system of declina- tion and hour angle is the axis of the equator, and the axis of ^i in the system of altitude and azimuth is the vertical line ; the angle between these axes is the complement of the latitude, or u = 90® — <p. Substituting this value of u in (44), and also the values of a:, y, ^, Xj, y,, ^„ given by (39) and (40), we have, after omitting the common factor J, Vol. L— 4 50 THE CELESTIAL SPHERE. COS li COS A=^9\n fp COS $ cos t — cos f sin H cos A sin -^ = cos d sin t sin h = cos fp cos ^ cos t + sin tp sin ^ which agree with (14). We see that when the element of dis- tance is left out of view (as it must necessarily be when the origin is not changed), the transformation by means of rectangu- lar co-ordinates leads to the same forms as the direct application of Spherical Trigonometry. . With regard to the entire generality of these formulae in their application to angles of all possible magnitudes, see Sph. Trig. Chap. IV. DIFFERENTIAL VARIATIONS OF CO-ORDINATES. 34. It is often necessary in practical astronomy to determine what effect given variations of the data will produce in the quan- tities computed from them. Where the formulre of computa- tion are derived directly from a spherical triangle, we can employ for this purpose the equations of finite diffvrcnccs [Sph. Trig. Chap. VI.] if we wish to obtain rigorously exact relations, or the simpler differential equations if the variations considered are extremely small. As the latter case is very frequent, I shall deduce here the most useful differential formuhe, assuming as well known the fundamental ones [Sph. Trig. Art. 153], da — cos C dh — cos B r/c trrr 8in 6 sin C d\ ^ — cos C da 4- dh — cos Kdc ^=z sin c sin A ^B > (46) — cos B tia — cos A db -\- dc — . Hin a sin B dC ) From these we obtain the following by eliminating da: sin db — cos a sin B dc ^^ sin b cos C d\ + sin a dB ) ,^-.x — cos rt sin C db + sin B dc —Hinc cos B d\ -f sin a dC ) and by eliminating db from these: sin a sin B dc = cos b d\ + cos a dB + dC (48) If we eliminate d\ from (47), we find cos b sin C db — cos c sin B dc -— sin c cos B dB — sin h cos C dC the terms of which being divided either by sin b sin C, or by its equivalent sin c sin H, we obtain cot b db — cot c dc =-- cot B (/B — cot C dC (49) DIFFERENTIALS OF CO-ORDINATES. 51 35. As an example, take the spherical triangle formed by the zenith, the pole, and a star, Art. 10, and put A = 180^ — ^ a = 90° — ^ C = ^ c = 90** — sp then the first equations of (46) and (47) give dd = — cos q dZ -{- sin q sin C dA -f- cos t d^ ^ x^qx cos d dt = sin q dZ -{- cos q sin C dA -f- sin i sin t d<p ) which determine the errors dd and dt in the values of d and t computed according to the formulae (4), (5), and (6), when (^, A^ and <p are affected by the small errors d^, dA, and d<p respectively. In a similar manner we obtain c?C = — cos q dd -{- sin q cos d dt -}- cos Ad^ 1 .;.j. sin C </^l = sin q dd -j- cos ^ cos d dt — cos C sin Ad<p ) which determine the errors rfj and dA in the values of f and A computed by (14), when 5, t, and ^ are aflfected by the small errors dd, dt, and dip respectively. 36. As a second example, take the triangle formed by the pole of the equator, the pole of the ecliptic, and a star. Art. 23. De- noting the angle at the star by tj, we find d^ = cos Tj dd — sin j) cos d da — ein X de ) .^«,. cos fi dX = sin y^ dd -{■ cos rj cos d da -{- sin fi cos X ds j v *-; and reciprocally, dd = cos ij di3 -f- sin rj cos /9 dX -[- sin a <7e ) ,-qv cos d da = — sin ly di3 -(- cos ly cos ^ c?>l — sin d cos a <fe J ^ 52 TiUE. CHAPTER 11. TIME USE OF TUE EPHEMERIS — INTERPOLATION — STAR CATALOGUES. 37. Transit. — The iiifltant when any point of the celestial nphere is on the meridian of an observer is designated as the tramii of tliat point over the meridian; also the meridian jxiasage^ and culmi)iaiio)i. In one complete revolution of the spliere about its axis, every point of it is twice on the meridian, at points which are 180° distant in right ascension. It is therefore necessarv to distin«:uish between the two transits. The meri- dian is bisected at the poles of the ccpuitor: the transit over that half of the meridian whi(»h contains the ol)scr\'er'8 zenith is the upper transit, or culmination; that over the half of the meri- dian which contains the nadir is the lower transit, or culmina- tion. At the upper transit of a point its hour angle is zero, or 0* ; at the lower transit, its hour angle is 12\ 38. The motion of the earth about its axis is perfectly uni- fona. If, then, the axis of the earth i)rescrved precisely the same direction in space, the ai»parcnt diurnal motion of the celestial sphere would also be perfectly uniform, and the inter- vals between the successive transits of any assumed point of the sphere would be perfectly equal. The effect of changes in the position of the earth's axis upon the transit of stai-s is most per- ceptible in the case of stars near the vanishing points of the axis, that is, near the poles of the heavens. AVe obtain a measure of time srn^ihf^ uniform by employing the successive transits of a point of the ecpnitor. The point most naturally indicated is the nrnal ctpunojr (also <*alled the Fii*st point of Aries, and de- noted by the symbol for Aries, T). 30. A siiUrenl ilmj is the interval of time between two succes- sive (upper) transits of the true vernal ecpiinox over the same meridian. The ettect of preiession and nutation upon the time of transit TIME. 53 of the vernal equinox is so nearly the same at two successive transits, that sidereal days thus defined are sensibly equal. (See Chapter XI. Art. 411.) The sidereal time at any instant is the hour angle of the vernal equinox at that instant, reckoned from the meridian westward from 0* to 24*. When TP is on the meridian, the sidereal time is 0* 0* 0* ; and this instant is sometimes called sidereal noon. 40. A solar day is the interval of time between two successive upper transits of the sun over the same meridian. The soUir time at any instant is the hour angle of the sun at that instant. In consequence of the earth's motion about the sun from west to east, the sun appears to have a like motion among the stars, or to be constantly increasing its right ascension ; and hence a solar day is longer than a sidereal day. 41. Apparent and mean solar time. — If the sun changed its right ascension uniformly, solar days, though not equal to sidereal days, would still be equal to each other. But the sun*8 motion in right ascension is not uniform, and this for two reasons : 1st. The sun does not move in the equator, but in the ecliptic, BO that, even were the sun's motion in the ecliptic uniform, its equal changes of longitude would not produce equal changes of right ascension; 2d. The sun's motion in the ecliptic is not uni- form. To obtam a uniform measure of time depending on the sun's motion, the following method is adopted. A fictitious sun, which we shall call the/r6'^ mean sun^ is supposed to move uniformly at such a rate as to return to the perigee at the same time with the true sun. Another fictitious sun, which we shall call the second mean sun (and which is often called simply the mean sun), is sup- posed to move uniformly in the equator at the same rate as the first mean sun in the ecliptic, and to return to the vernal equinox at the same time with it. Then the time denoted by this second mean sun is perfectly uniform in its increase, and is called mean time. The time which is denoted by the true sun is called the tnie or, more commonly, the apparent time. The instant of transit of the true sun is called apparent noon, and the instant of transit of the second mean sun is called ynean noon. 4 'A TIME. Tlie (quation of time is the difFerence between apparent and r.'j'.an time ; or, in other words, it is the difterence between the hour angles of the trne sun and the second mean sun. The ;rn:ate.st difFerence is about IG" The equation of time is also the difterence between the right a-''*;iiHioiis of the true sun and the second mean sun. The right a-rension of the second mean sun is, according to the preceding 'h'fmitionH, equal to the longitude of the liret mean sun, or, as it i- 'ommonly called, the sun*s mean longitude. To compute the «'qiiatioii of time, therefore, we must know how to iind the longi- VuU: of the first mean sun ; and this'is deduced from a knowledge of the true sun's apparent motion in the ecliptic, which belongs to Phyrti<;al Astronomy. Here it suffices us that its value 18 iriven for each day of the year in the Ephemeris, or Nautical Ahnunuc. 42. Asfronomkal time. — The solar day (apparent or mean) is concreived by astronomers to commence at noon (apparent or nn'un), and is divided into twenty-four hours, numbered succes- nivclv from to 24. Astronomical time (apparent or mean) is, then, the hour angle of the sun (apjmrent or mean), reckoned on the equator xce^t- ward throughout its entire circumference from 0* to 24*. 43. Cirll time. — For the conmion pur[)oses of life, it is more convenient to begin the day at midnight, that is, when the sun is on the meridian at its lower transit The civil (hiy is divided into two periods of twelve hours each, namely, from midnight to noon, marked A.M. (Ante Meridiem), and from noon to midnight, marked P.M. (Post Meridiem) 44. T(i convert eiril into astronomical tirne, — The civil day begins 12'^ before the astronomical day of the same date. This remark is the only precept that need be given for the conversion of one of these kinds of time into the other. Examples. Ast. T. May 10, ir)*=_r Civ. T May 11. 3* A.M. Ast. T. Jan. 3, 7*.-: Civ. T. Jan. 8, 7* P.M. Ast. T. Aug 31, 20* r-_ Civ. T. Sept. 1, 8» A.M. TIME. 55 45. Time at different meridians. — The hour angle of the sun at any meridian is called the local (solar) time at that meridian. The hour angle of the sun at the Greenwich meridian at the same instant is the corresponding Greenwich time. This time we shall have constant occasion to use, both because longitudes in this country and England are reckoned from the Greenwich meridian, and because the American and British Nautical Almanacs are computed for Greenwich time.* The* difference between the local time at any meridian and the Greenwich time is equal to the longitude of that meridian from Greenwich, expressed in time, observing that 1* =^ 15°. The difterence between the local times of any tvvo ^ig- ^o- meridians is equal to the difterence of longitude of those meridians. In comparing the corresponding times at two dif- ferent meridians, the most easterly meridian may be distinguished as that at which the time is greatest ; that is, latest. ^ m a m^ If then PM^ Fig. 10, is any meridian (referred to the celestial sphere), PG the Greenwich meridian, PS the declination circle tlirough the sun, and if we put !r„ = the Greenwich time = GPS, T =z the local time = MPS, L = the west longitude of the meridian PM = GPMj we have T^=T^L I y^^^ If the given meridian were east of Greenwich, as PJM^', we should have its east longitude = T — T^\ but we prefer to use the general formula Z/= T^— T in all cases, observing that east longitudes are to be regarded as negative. In the formula (54), T^ and T are supposed to be reckoned always westward from their respective meridians, and from 0* to 24* ; that is, T^ and T are the astronomical timeSy which should, of course, be used in all astronomical computations. As in almost every computation of practical astronomy we are dependent for some of the data upon the ephemcris, — and these ♦ What we have to say respecting the Greenwich time is, however, equally appli- cable to the time at any other meridian for which the ephemeris may be computed. 56 TIME. are commonly given for Greenwich, — it is generally the first step in such a computation to deduce an exact or, at least, an ap- proximate value of the Greenwich astronomical time. It need hardly be added that the Greenwich time should never be other- wise expressed than astronomically.* Examples. 1. In Longitude 76° 32' W. the local time is 1856 April 1, 9* 3'^ lO- A.M. ; what is the Greenwich time ? Local Ast. T. March 31, 21» 3- 10- Longitude +56 8 Greenwich T. -Aprin^ 2 9 18 2. In Long. 105° 15' E. the local time is August 21, 4* 3" P.M ; what is the Greenwich time ? Local Ast. T. Aug. 21, 4» 3* Longitude — 7 1 Greenwich T. Aug. 20, 21 2 3. Long. 175° 30' W. Loc. T. Sept. 30, 8* 10- A.M. = G. T. Sept. 30, 7* 52-. 4. Long. 165° 0' E. Loc. T. Feb. 1, 7* 11- T.M. = G. T. Jan. 31, 20* 11-. 5. Long. 64° 30' E. Loc. T. June 1, 0* M. (Noon) n= G. T. May 31, 19* 42-. 46. In nautical practice the obsers'cr is provided with a chro- nometer which is regulated to Greenwich time, before sailing, at a phice wliose longitude is well known. Its error on Green- wich time is carefully determined, as well as its daily gain or loss, that is, its rafc, so that at any subsecpient time the Green- wich time niavbe known from the indication of the chronometer corrected for its error and the accumulated rate since the date of sailing. As, however, the chronometer has usually only 12* marked <m the dial, it is necessary to distinguish whether it indicates A.M. or I\M. at (ireenwich. This is ahvavs readilv done by means of the observer's approximate longitude and local * On this account, chronometern intended for nautical and astronomical purposet Mhould always be marked from <** to 24*. instead of from (»* to 12* as is now usual. It id 8urj>ri!«ing that navigators have not insisted upon this point. TIME. 57 time. As this is a daily operation at sea, it may be well to illus- trate it by a few examples. 1. In the approximate longitude 5* W. about S^ P.M. on Au- gust 3, the Greenwich Chronometer marks 8* ll*" 7*, and is fast on G. T. 6"* 10' ; what is the Greenwich astronomical time ? Approx. Local T. Aug. 3, 3» Gr. Chronom. 8* ll* 7* " Longitude, + 5 Correction, — 6 10 Approx. G. T. Aug. 3, 8 Gr. Ast. T. Aug. 3, 8 4 57 2. Li Long. 10* E. about 1* A.M. on Dec. 7, the Greenwich Chronometer marks 3* 14"* 13'.5, and is fast 25** 18'.7 ; what is theG. T.? Approx. Local T. Dec. 6, 13» Gr. Chronom. 3* 14- 13'.5 <* Long. — 10 Correction, — 25 18 .7 Approx. G. T. Dec. 6, 3^ G. A. T. Dec. 6, 2 48 54.8 3. In Long. 9* 12" W. about 2* A.M. on Feb. 13, the Gr. Chron. marks 10* 27"* 13*.3, and is slow 30- 30^.3; what is the G. T.? Approx. Local T. Feb. 12, 14* Gr. Chronom. 10» 37- 13'.3 " Long. +9_ Correction, + 30 30.3 Approx. G. T. Feb. 12, ~23^ G. A. T. Feb. 12, 23 7 43-6 The computation of the approxim'ate Greenwich time may, of course, be performed mentally. 47. The formula (54), i= 7;— T, is true not only when T^ and T are solar times, but also when they are any kinds of time whatever, or, in general, when 7J, and 7^ express the hour angles of any point whatever of the sphere at the two meridians whose difterence of longitude is L. This is evident from Fig. 10, where S may be any point of the sphere. 48. To convert the apparent time at a given meridian into the mean iimCj or the mean into the apparent time. If M = the mean time, A = the corresponding apparent time, E = the equation of time, we have M=A + lf! or A =M--E 08 TIME. in which E is to l>c regarded as a positive quantity when it is adffifici to ajfpurent t'uiu\ The vahie of E is to be taken from the Xautical Ahuanae for the Greenwich instant corresponding to the given h)cal time. If apparent time is given, find the Gr. apparent time and take E from page I of the month in the Xautical Ahuanae; if mean time is given, find the Gr. mean time and take E from page II of the month. Example 1.— In h^ngitude 60° W., 1856 May 24, 3* 12* IQr P.M., apparent time; what is the mean time? We have first Local time May 2i, 3* 12- 10' Longitude, 4 Gr. app. time May 24, 7 12 10 We must, therefore, find E for the Gr. time. May 24, 7* 12* 10', or 7\21. By the Xautical Almanac for 18r)6, we have ^at ajiparcnt (ireenwich noon May 24 - - 8"*25'.43, and the hourly dittercnce -j- 0'.224. Hence at the given time /; :=.. — 3- 2;V.43 + 0'.224 X 7.21 =:= — 3- 23*.81 and the recpiired mean time is M. - 3* 12- 10- — 3- 23'.«1 ~. 3* 8- 40'. 19. Example 2.— In longitude 60° W., 18;56 May 24, 3*8-46M9 mean time; what is the api>arcnt time? (Jr. mean time. May 24, 7* S<- 46-11) ( - 7M5) /; at mean noon May 24 =-- — 3- 2;V.41 Hourly diff. — 0'.224 Correction for 7*. IT) — 4- 1 .r»0 7.15 E=^— 3 23.S1 im and hence — A'- :+ 3 2:i .HI A -^:i 12 10 .00 As tlM* equation of time is not a uniformly varying quantity, it is not (juite arcuratc to eomiJUtc its correction as above, hy mnl- tii»Ivinir the ixiviMi honrlv <liffercncc hv the numher of hours in the (Jreenwieli tinn*. ft»r tliat ]»rocess assumes that this hourly differen«M' is the same for eaeh hour. The variations in the h«»urly ditlerence are, however, so small that it is oidy when TIME. 59 extreme precision is required that recourse must be had to the more exact method of interpolation which will be given here- after. 49. To determine the relative length of the solar and sidereal \inits of time. According to Bessel, the length of the tropical year (which is the interval between two successive passages of the sun through the mean vernal equinox) is 365.24222 mean solar days;* and since in this time the mean sun has described the whole arc of the equator included between the two positions of the equinox, it has made one transit less over any given meridian than the vernal equinox ; so that we have 366.24222 sidereal days = 365.24222 mean solar days whence we deduce 1 Bid. day = ^^^'^^^^^ sol. day =' 0.99726957 sol. day ^ 366.24222 ^ ^ or 24* sid. time = 23* 56* 4-.091 solar time Also, 1 sol. day = ^^j^ gi^. day = 1.00273791 sid. day ^ 365.24222 ^ ^ or 24* sol. time == 24* 3- 56«.555 sid. time If we put 366.24222 ^^ 365.24222 and denote by /an interval of mean solar time, by /' the equiva- lent mterval of sidereal time, we always have /' = /i7 = /-I- 0£ — 1) 7 =7+ .00273791 7 \ I =i^ =r^n — l)r = 7' — .00273043 r } ^^^^ ft ^ fi^ ) Tables are given in the Nautical Almanacs to save the labor of computing these equations. In some of these tables, for each solar inter\^al 7 there is given the equivalent sidereal interval I' = fiT, and reciprocally: in others there are given the correc- tion to be added to 7 to find 7' {i.e. the correction .00273791 7), * The length of the tropical year is not absolutely constant. The value given in the text is for the year 1800. Its decrease in 100 years is about C'.6 (Art. 407). CO TIME. and the correction to he subtracted from /' to find / (i.e. the correction .00273043 /'). The latter form is the most conve- nient, and is adopted in the American Ephemeris. The correction (/i — 1) 7 is frequently called the accckration of the fixed stars (re- latively to the sun). The daily acceleration is 3"* 56'.665. 50. To convert the mean solar time at a givai meridian into the corresponding sidereal time. In Fig. 1, page 25, if PQ is the given meridian, VQthe equator, D the mean sun, Vthe vernal equinox, and if we put T= I)Q=i the moan solar time, 0= VQ = the sidereal lime, := the right ascension of the meridian, V= the right ascension of tho~mean sun, we have e==T+V (56) The right ascension of the mean sun, or V, is given in the American Ephemeris, on page II of the month, for each Green- wich mean noon. It is, however, there called the " Sidereal Time,*' because at mean noon the second mean stin is on the meridian, and its right ascension is also the right ascension of the meridian, or the sidereal time. But this quantity V is uni- formly increasing* at the rate of 3"* 56'.555 in 24 mean solar hours, or of i>*.8565 in one mean hour. To find its value at the given time T, we may first find the Greenwich mean time T^ by applying the longitude ; then, if we put T'i, ^n the value of V at Gr. mean noon, = the " sidereal tinio" in the ephemeris for the given date, we have in which T^ must be expressed in hours and decimal parts. It is easilv seen that 1>'.8565 is the acceleration of sidereal time on solar time in one solar hour, and therefore the term I>'.8.5G5 X 7^ is the correction to add to 7{,t() reduce it from a solar to a side- real interval. This term is identical with (/i — 1)7)^ as given by * The Midcrenl time at mean noon is equal to the tntf R.A. of the mean sun, or it is the R.A. of the mean sun referred to the truf equinox, ami therefore inToWes the nutation, so that its rate of increase is not, strictly, uniform. But it is sufBcientlj m for 24 hours to be so regarded in all practical computations. See Chapter XI. TIME. 61 tlie preceding article, if T^ in the latter expression is expressed in seconds, since wc have -?^:?^ :- 0.0027391= /I -1 3600* "We may then write (56) in the following form, putting L = the west lon^tude of the given meridian, and T^J= T+ L: e=!r+r, + (/x-i)(r+i) (57) The term (ji — 1) {T + L) is given in the tables of the Amer- ican Ephemeris for converting "Mean into Sidereal Time,*' and may be found by entering the table with the argument T + A or by entering successively with the arguments T and L and adding the corrections found, observing to give the correction for the longitude the negative sign when the longitude is east. If no tables are at hand, the direct computation of this term will be more convenient under the form 9".8565 X ^^o- Example 1.— In Longitude 165° W. 1856 May 17, 4* A.M.; what is the sidereal time ? The Greenwich time is May 17, 3*; and the computation may be arranged as follows : Local Ast. Time T == 10» 0- 0*. At Gr. Noon May 17, Vo= 3 41 28 .32 Correction of V^ for 3* ) <^q ;.- = 9*.8565 X 3 e= 19 41 57.89 Example 2.— Li Longitude 25° 17' E. 1856 March 13, about 9* 30* P.M., an observation is noted by a Greenwich chronometer which gives 7* 51"* 12'.3 and is slow 3"* 13\4 ; what is the local sidereal time ? Gr. mean date, March 13, 7* 54* 25*.7 Longitude, 1 41 8 E. T = 9~~35~33T March 13, ro=23 25 12 .26 Tabular corr. for 7* 54- 25v7 = 1 17 .94 e =~9 2 3T90 C2 TIME. Example 3.— In Longitude 7* 25" 12- E. 1856 March 13, 18* 15- 47'.3 mean local astronomical time ; what is the sidereal time? T =13»15-47'.3 F„=23 25 12.26 Tabular corr. for 13* 15- 47v3 = + 2 10 .73 Tab. corr. for long. — 7* 25- 12-. = ~ 1 13.14 e = 12 41 57 .15 51. To convert the apparent solar time at a given meridian into the sidereal time at that meridian. Find the mean time by Art. 48, and then the sidereal time by Art. 50. Or, more directly, to the given apparmt time add the true sun*s right ascension. For if in Fig. 1 we take D as the true sun, we have DQ ^= apparent solar time, VD = R. A. of true sun, and VQ, the sidereal time, is the sum of these two. The right ascension of the true sun is called in the Ephemeris the "sun's apparent right ascension,'* and is there given for each apparent noon. It is not a uniformly increasing quantity; but for many puq^oses it will be sufficiently accurate to consider the hourly increase given in the Ephemeris as constant for 24*, and to add to the app. K. A. of the Ephemeris the correction found by multiplying tlie hourly difterence by the number of hours in the Greenwich time. Example.— In Longitude 98° W. 1856 June 3, 4* 10« P.M. app. time ; what is the sidereal time ? (ir. app. date June 8, 10* 42- (^ 10*.7) Local app. t. — 4* 10- 0*. O's App. K. A. App. noon June 3 = 4 46 22 .04 Hourly diff. = 10'.271 Corr. :^ 10\271 X 10.7 = 1 49 .9 Sidereal time =8 58 11 .94 52. To conrcrt the sidereal time at a given meridian into the mean time at that meridian. First ynethod. — AVhen the Greenwich mean time is also given, as is fro(picntly the case, we have (»nly to find V as in Art. 50 by a<Ming to Vj, given in the Ephemeris the correction for the (ireenwich time taken from the table ''Mean into Sidereal Time," and then we have, by transposing cipiation (56), T :- e - V TIME. 63 Example. — In Longitude 165° W., the Greenwich mean time being 1856 May 17, 3*, the local sidereal time 19* 41*^ 57*.89, what is the local mean time ? Fo = 3» 41- 28-.32 Corr. for 3* r= + 29.57 r = 3 41 57 .89 e =z 19 41 57 .89 6 — F= T = 16 0.00 The longitude being 11* W., the local date is May 16. Second method. — Wlien the Greenwich mean time is not given, we can find T from (57), all the other quantities in that equation being known. Wo find or, in a more convenient form for use, r=e-F.-(i-l)(e-F. + i) (58) in which the term multiplied by 1 — — is the retardation of mean time on sidereal in the interval © — l^o + ^^y ^^^^^ '^^ given in the table "Sidereal into Mean Time." It is convenient to enter the table first with the argument © — V^ and then with the argu- ment //, and to subtract the two corrections from © — V^^ ob- serving that the correction for the longitude becomes additive if the longitude is east. Example.— In Longitude 165° W. 1856 May 16, the sidereal time is 19* 41"* 57'.89; what is the mean local time? e = 19* 41-» 57v89 May 16, Y^ = 3 ^7_3K76 e — T; =T6 ~T~26~13 Table, "Sidereal into | Corr. for 16* 4- 26'.13 = — 2 38 .00 Mean Time" I " " longitude 11* == _ 1 48 .13 r="i6 oToo 63. The following method of converting the sidereal into the mean time is preferred by some. In the last column of page III of the month in the American Naut. Aim. is given the **Mean Time of Sidereal 0*.*' This quantity, which we may denote by F', is the number of hours the mean sun is west of the vernal 64 TIME. equinox, and is merely the difterence between 24* and the mean Bun*8 right ascension. The hour angle of the mean sun at any instant is then the hour angle of the vernal equinox increased hy the value of F' at that instant. To find this value of V% we first reduce the Almanac value to the given meridian by cor- recting it for the longitude by the table for converting sidereal into mean time; then reduce it to the given sidereal time © (which is the elapsed sidereal time since the transit of the vernal equinox over the given meridian) by further correcting it by the same table for this time 0. We then have the mean time 7^ by the formula It is necessary to observe, however, that if + V'' exceed 24* it will increase our date by one day; and in that case V should be taken from the Ahnanac for a date one day less than the given date; that* is, we must in every case take that value which belongs to the Greenwich transit of the vernal equinox immediately j^rctu^dinf/ that over the given meridian. Example. — Same as in Art. 52. e = 19* 41- 57*.89 May 15, TV = 20 23 3 .88 Corr. for long. 11* W. ..:. — 1 48 .13 Corr. for 19* 41" 58- = —3 13 .04 T^='i0 ~0 0.00 54. To find the hour angk of a star* at a given tunc at a given meridian. In Fig. 1, we have for the star at 0, DQ -- VQ — VD\ that is, if wc put the sidereal time, a . the right ascension of the star, t =--. the hour angle << a. a then t -, e - - a (59) If a exceeds 0, this tormula will give a negative value of i which will express tlu* hour angle east of the meridian: in that case, if we increa.«<e bv 24* before subtractintc Gt, we shall find ♦ W<? shall uxe "star," for brevity, to denote any cck'!<tiul hody. HOUR ANGLES. 65 the value of t reckoned in the usual manner, west of the meri- dian. According to this formula, then, we have first to convert the given time into the sidereal time, from which we then subtract the right ascension of the star, increasing the sidereal time by 24* when necessary; the remainder is the required hour angle west of the meridian. In the case of the sun, however, the apparent time is at once the required hour angle, and we only have to apply to the given mean time the equation of time. Example.— In Longitude 165^ W. 1856 May 16, 16* 0« O' mean time, find the hour angles of the sun, the moon, Jupiter, and the star Fomalhaut. The Greenwich mean date is 1856 May 17, 3*, and the local sidereal time is (see Example 1, Art. 50) = 19* 41- 57'.89. For the Greenwich date we find from the Naut. Aim. the equa- tion of time Ey and the right ascensions a of the moon, Jupiter, and Fomalhaut, as below : T = 16* 0* 0* e = 19* 41* 57*.89 — ^= + 3 49.85 y8a = 13 50 21.35 O's ^ = 16 3 49 .85 e = 19* 41- 57*.89 Ql's a = 7 57.52 Ql's t = 19 34 .37 Fomalh. ^ = 20 52 17 .71 K the sidereal time had been given at first, we should have found the hour angle of the sun by subtracting its apparent right ascension as in the case of any other body. 55. Given the hour angle of a star at a given meridian on a given day J to find the local mean time. By transposing the formula (59), we have e = < + a ' (60) so that, the right ascension of the star being given, we have only to add it to the given hour angle to obtain the local sidereal time, whence the mean time is found by Art. 52. When the sum t + a exceeds 24*, we must, of course, deduct 24*. If the body is the sun, however, the given hour angle is at once the apparent time, whence the mean time as before. But if the body is the moon Vol. L— 6 5'8 t — 5 51 36.54 O — 19* 41- 57'.89 Fomalh. a 22 49 40.18 66 TIME. or a planet, its right ascension can be found from the Ephemeris only when we know the Greenwich time. If then the Green- wich time is not given, we must find an approximate value of the local time by formula (60), using for a a value taken for a Greenwich time as nearly estimated as possible ; from this local time deduce a more exact value of the Greenwich time, with which a more exact value of a may be found ; and so repeating as often as maybe necessary to reach the required degree of precision. Example 1. — In Longitude 165® W. 1856 May 16, the hour angle of Fomalhaut is 20* 52-* 17'.71 ; what is the mean time ? t = 20* 52* 17'.71 May 16, Fomalh. a = 22 49 40 .18 e = 19 41 57 .89 whence the mean time is found to be T= 16* 0* 0*. Example 2. — In Longitude 165° W. 1856 May 16, the moon's hour angle is 5* 51* 36*.54, and the Greenwich date is given May 17, 3* ; what is the mean time ? t = 5» 51- 36*.54 For May 17, 3», a = 13 50 21.35 6 = 19 41 57.89 " Mayl7, 3\ F= 3 41 57.89 T = 16 .00 Example 3. — In Longitude 30° E. 1856 August 10, the moon's hour angle is 4* 10* 53*.2; what is the mean time ? For a first approximation, we ob8er\'e that the moon passes the meridian on August 10 at about 7* mean time (Am. Eph. page IV of the month), and when it is west of the meridian 4* the mean time is about 4* later, or 11*, from which subtracting the longitude 2* we have, as a rough value of the Greenwich time Aug. 10, 9*. We then have t — 4» 11« For Aug. 10, 9*, a = 16 29 6 — 20 40 " Aug. 10, 9*, F— 9 18 Ist approx. value 7= 11 22 Hence the more exact Greenwich date is Aug. 10, 9* 22*; and with this we now repeat : HOUR ANGLES. 67 t= 4*10*53-.2 For Aug. 10, 9» 22- a = 16 29 26.8 6 = 20 40 20.0 »' « V= 9 18 8.1 2d approx. valud T = 11 22 11 .9 A third approximation, setting out from this value of T, gives us r= 11* 22- 12'.32. 56. The mean time of the meridian passage not only of the moon but of each of the planets is given in the Ephemeris. This quantity is nothing more than the arc of the equator in- tercepted between the mean sun and the moon's or planet's declination circle. If we denote it by M^ we may regard J!f as the equation between mean time and the lunar or planetary time, these terms being used instead of "hour angle of the moon" or "hour angle of a planet," just as we use "solar time" to signify "hour angle of the sun." This quantity M is given in the Ephe- meris for the instant when the lunar or planetary time is 0*, and its variation in 1* of such time is also given in the adjacent column. If, then, when the moon's or a planet's hour angle at a given meridian = tj we take out from the Almanac the value of M for the corresponding Qreen^vich value of /, we shall find the mean time jTby simply adding Mio t; that is, T=t+M (61) This is, in fact, the direct solution of the problem of the pre- ceding article, and neither requires a previous knowledge of the Greenwich mean time nor introduces the sidereal time. But the Almanac values of M are not given to seconds ; and there- fore we can use (61) only for making our first approximation to Tj after which we proceed as in the last article. The Green- wich value of t with which we take out M is equal to < + i, denoting by L the longitude of the given meridian (to be taken with the negative sign when east), and the required value of M is the Almanac value increased by the hourly diff*. multiplied by {t + L) in hours. As the hourly diff. of M in the case of the moon is itself variable, we should use that value of it which corresponds to the middle of the interval t-\-L; that is, we should first correct the hourly diff. by the product of its hourly change into J (< + i) in hours. 68 EPHEMERIS. Example. — Same as Example 3, Art. 55. We have f 4. X = 2* 10- 53'.2 = 2M8 t = 4» 10- 53'.2 At Gr. trans. Hour. Diff. =2".17 AtGr.traTis.Aug.lO,i/^=: 7 6 30 Variation of H. D. in 1* 5- = .01 2-.18 X 2.18 = + 4 45 Corrected Hourly Diff. = 2 .18 T= 11 22 8 .2 which agrees within 4* with the true value. Taking it as a first approximation, and proceeding as in Art. 55, a second approxima- tion gives T = 11* 22^ 12M9. THE BPHEMERIS, OR NAUTICAL ALMANAC. 57. We have already had occasion to refer to the Ephemeris ; but we propose here to treat more particularly of its arrange- ment and use. The Astronomical Ephemeris expresses in numbers the actual state of the celestial sphere at given instants of time ; that is, it gives for such instants the numerical values of the co-ordi- nates of the principal celestial bodies, referred to circles whose positions are independent of the diurnal motion of the earth, as declination and right ascension, latitude and longitude; together with the elements of position of the circles of re- ference themselves. It also gives the eftects of changes of posi- tion of the observer upon the co-ordinates, or, rather, numbers from which such changes can be readily computed (namely, the parallax, which will be fully considered hereafter), the ap- parent angular magnitude of the sun, moon, and planets, and, in general, all those phenomena which depend on the time; that is, which may be regarded simply Rsfimr(io)}s of (he time. The Americav Ephemeris is composed of tsvo parts, the first computed for the meridian of Greenwich, in conformity with the British Nautical Almanac, especially for the use of navigators ; the second computed for the meridian of Washington for the convenience of American astronomers. The French Ephemeris, La Connaissance des Tanps, is computed for the meridian of Paris; the German, Berliner Astronomisches Jahrbuch, for the meridian of Berlin. All these works are published annually several years in advance. 58. In what follows, we assume the Ephemeris to be computed for the Greenwich meridian, and, consequently, that it contains the right ascensions, declinations, equation of time, &c. for given equidistant instants of Greenwich time. EPHEMERIS. 69 Before we can find from it the values of any of these quanti- ties for a given local time, we must find the corresponding Green- wich time (Arts. 45, 46). When this time is exactly one of the instants for which the required quantity is put down in the Ephe- meris, nothing more is necessary than to transcribe the quantitj- as there put down. But when, as is mostly the case, the time falls between two of the times in the Ephemeris, we must obtain the required quantity by interpolation. To facilitate this inter- polation, the Ephemeris contains the rate of change, or difterence of each of the quantities in some unit of time. To use the difference columns with advantage, the Qreenwnch time should be expressed in that unit of time for which the difference is given : thus, when the difference is for one hour, our time must be expressed in hours and decimal parts of an hour ; when the difference is for one minute, the time should be expressed in minutes and decimal parts, &c. 69. Simple interpolation, — In the greater number of cases in practice, it is sufficiently exact to obtain the required quantities by simple interpolation ; that is, by assuming that the differences of the quantities are proportional to the differences of the times, which is equivalent to assuming that the differences given in the Ephemeris are constant. This, however, is never the case; but the error arising from the assumption will be smaller the less the interval between the times in the Ephemeris ; hence, those quantities^ which vary most irregularly, as the moon's right ascension and declination, are given for every hour of Green- wich time ; others, as the moon's parallax and semidiameter, for every twelfth hour, or for noon and midnight ; others, as the sun's right ascension, &c., for each noon ; others, as the right ascensions and declinations of the fixed stars, for every tenth day of the year. Thus, for example, the greatest errors in the right ascensions and declinations found from the American Ephe- meris by simple interpolation are nearly as follows : — Error in R. A. Error In Decl. Sun C.l 3".5 Moon 0.1 1 .5 Jupiter 0.1 .6 "Mars 0.4 2 .4 Venus 0.2 5 .4 70 EPHEMEBI8. To illuBti^ate simple interpolation when the Greenwich time is given, we add the following Examples. For the Greenwich mean time 1856 March 30, 17* 11" 12*, find the following quantities from the American Ephemeris : thQ Equation of time, the Right Ascension, Declination, Hori- zontal Parallax, and Semidiameter of the Sun, the Moon, and Jupiter. 1. The Equation of time,— The Gr. T. = March 30, 17* 11-2 = March 30, 17M87. (Page II) E at mean noon = + 4- 27*.ll H. D. = — 0'.763 Corr. for 17M9 = — 13.11 17.19 ^ = _|_ 4 U .00 — 13.11 Note. — Observe to mark E always with the sign which denotes how it is to be applied to apparent time. If increafingy the II. D. (hourly difference) should hare the same sign as E; otherwise, the contrary sign. 2. SurCs E, A. and Dec, (P. II.) a at 0* = 0* 36- 40'.78 H. D. + 9*.094 Corr. for 17M87 = + 2 36.29 17.1^7 a= 39 17.07 156.29 ^ at 0* = + 3° 67' 21".9 H. D. + 58'M5 Corr. for 17M87 = + 16 39 .4 17.187 ^ = _^ 4 14 1 .3 999.4 3. Moon's E, A, and Dec, a at 17* = 20* 18-' ^M Diff. 1- + 2'.4975 Corr. for 11*2 = + 27 .97 11.2 a= 20 18 37 .77 27.97 d at 17* = — 25^ 3' 10".9 Diff. 1- + 8".275 Corr. for 11-.2 = + 1 32 .7 11.2 ^ = — 25 1 38 .2 92.68 4. Moon's Hor. Par, (= r) and Semid. (= S). n at 12* = 58' 44".l H. D. + 2".17 Corr. for 5*.2 = + 11 .3 5^ r = 58 55 .4 11.28 EPHEMERIS. 71 S at 12* = 16' 2".0 Diff. in 12* = + 7'M Corr. for 5*.2 = + 3 .1 5=16 5 .1 5. Jupiter's R, A. and Dec, a at 0* = 23* 29- 49'.95 H. D. + 2'.175 Corr. for 17M87 = + 37.38 17.187 a = 23 30 27.33 37.38 ^ at 0* = — 4« 22' 45".6 H. D. + 13".74 Corr. for 17*.187 = + 3 56 .1 17.187 ^ = — 4 18 49 .5 236.1 6. Jupiter's Hor, Par. and Semid. — At the bottom of page 231, we find for the nearest date March 31, without interpolation : It = 1".5 S = 15".7 NoTK. — It maj be obserred that we mark hourlj differences of declination plut, when the body is moving northward, and minus when it is moving southward, m In the above we have carried the computation to the utmost degree of precision ever necessary in simple interpolation. 60. To find the right ascension and declination of the sun at the time of its transit over a given meridian^ and also the equation of time at the same instant. When the sun is on a meridian in west longitude, the Green- wich apparent time is precisely equal to the longitude, that is, the Gr. App. T. is after the noon of the same date with the local date, by a number of hours equal to the longitude. When the sun is on a meridian in east longitude, the Gr. App. T. is before the noon of the same date as the local date, by a number of hours equal to the longitude. Hence, to obtain the sun's right ascension and declination and the equation of time for apparent noon at any meridian, take these quantities from the Ephemeris (page I of the month) for Greenwich Apparent Noon of the same date as the local date, and apply a correction equal to the hourly difference multiplied by the number of hours in the lon- ^tude, observing to add or subtract this correction, according as the numbers in the Ephemeris may indicate, for a time before or after noon. 72 EPHBMERIS. Example 1.— Longitude 167° 31' W. 1856 March 20, App. Noou, find O's R. A., O's Dec, and Eq. of T. Longitude = + 11* 10» 4* = + 11M7 a at App. 0* = 0* 0* 20'.94 H. D. + O'.OOS Corr. for + 11M7 = + 1 41.62 + 11.17 a= 2 2.56 + 101.62 d at App. 0*= + 0° 2' 16".5 H. D. + 59".21 Corr. for + 11M7 = + 11 1 .4 -f 11.17 ^ = -f- 13 17 .9 + 661.4 E at App. 0* = + 7- 31*.57 H. D. — 0'.759 Corr. for + 11M7 = — 8.48 + 11.17 E = -{- 7 23 .09 — 848 Example 2.— Longitude 167° 31' E. 1856 March 20, App. Noon, find O's R.A., 0*8 Dec, and Eq. of T. Longitude = -— 11* 10- 4*= — 11M7 a at App. 0* = 0* 0- 20*.94 H. D. + 9*.098 Corr. for — 11M7 = — 1 41 .62 -Z_^^'yL a = 23 58 39 .32 -^01j62 d at App. 0* = + 0° 2' 16".5 H. D. + 59".21 Corr. for — 11M7 = — 11 1 .4 — 11.17 * d — 8 44 .9 — 661.4 E at App. 0* — + 7- 31*.67 M. D. 0*.759 Corr. for — 11*. 17 — + 8.48 11.17 J^ = + 7 40 .05 + 8.48 61. To find the mean local time of the moon's or a planefs transit over a givai meridian. This ift the flame as the problem of Art. 55, in the Bpecial case where the hour angle of the moon or planet at the given meri- dian is 0*. We can, however, obtain the required time directly from the Ephemeris, with sufficient accuracy for many purposes, * In this example the sun crosses the equator between the times of its transits OTer the local and the Greenwich meridians. The case must he noted, as it is a f^- quent occasion of error among navigators. The same case can occur on September 22 or 28. EPHEMBRIS. 73 by simple interpolation. On page IV of the month {Am. Ephem. and British NauU Aim.) we find the mean time of transit of the moon over the Greenwich meridian on each day. This mean time is nothing more than the hour angle of the mean sun at the instant, or the difference of the right ascensions of the moon and the mean sun ; and if this difference did not change, the mean local time of moon's transit would be the same for all meridians; but as the moon's right ascension increases more rapidly than the sun's, the moon is apparently retarded from transit to transit. The difference between two successive times of transit given in the Ephemeris is the retardation of the moon in passing over 24* of longitude, and the hourly difference given is the retardation in passing from the Greenwich meridian to the meridian 1* from that of Greenwich. Hence, to find the local time of the moon's transit on a given day, take the time of meridian passage from the Ephemeris for the same date (astro- nomical account) and apply a correction equal to the hourly difference multiplied by the longitude in hours; adding the correction when the longitude is west, subtracting it when east. The same method applies to planets whose mean times of transit are given in the Ephemeris as in the case of the moon. Example.— Longitude 130° 25' E. 1856 March 22 ; required local time of moon's transit. Gr. Merid. Passage March 22, 13*. 2«.7 H. D. + V^.m Corr. for Long. — 8*.7 = — 13.8 — 8J Local M. T. of transit = 12 48.9 — 13.8 62. To find the moon's or a planefs right ascension, declination, ^c.y at the time of transit over a given meridian. Find the local time of transit by the preceding article, deduce the Greenwich time, and take out the required quantities from the Ephemeris for this time. This is the usual nautical method, and is accurate enough even for the moon, as meridian observa- tions of the moon at sea are not susceptible of great precision. For greater precision, find the local time by Art. 55 for t = 0*, and thence the Greenwich time. See also Moon Culminations, Chapter VU. 63. Interpolation by second differences. — The differences between the successive values of the quantities given in the 74 EPHEMERIS. Ephemeris as functions of the time, are called the first differ- mccs; the differences between these successive differences are called the second differences; the differences of the second differ- ences are called the third differences^ &c. In simple interpolation we assume the function to vary uniformly ; that is, we regard the first difference as constant, neglecting the second difference, which is, consequently, assumed to be zero. In interpolation by second differences we take into account the variation in the first difference, but we assume its variations to be constant; that is, we assume the second difterences to be constant and the third differences to be zero. AVTien the American Ephemeris is employed, we can take the second difterences into account in a very simple manner. In this work, the difterence given for a unit of time is always the difterence belonging to tlie instant of Greenwich time against which it stands, and it expresses, therefore, the rate at which the function is changing at that instant. This difterence, which we may here call the first difterence, varies with the Greenwich time, and (the second difterence being constant) it varies uni- formly, so that its value for any intermediate time may be found by simple inteq^olation, using the second difterences as first dif- ferences. Now, in computing a correction for a given interval of Greenwich time, we should employ the meany or average value, of the first difterence for the interval, and this mean value, when we regard the second differences as constant, is that which belongs to the middle of the interval. Hence, to take into account the second difterences, we have only to obser\'e the very simple rule — employ that (interpolated) value of the first difference which corresponds to the middle of the interval for ichich the correction is to be computed. Example. — For the Greenwich time 1856 March 2, 12* 29* 36% find the moon's declination. March 2. V2^{A) :^ — 27° 10'41".8 DiflF. 1- .^ -f 4".814 2d Diff. = + (r.l89 Corr. for 29'».6 -f 2 23 .9 Corr. for2dd iff. -f .047 0.26 <J= —27 8 17 .9 -f 4.861 -f 0.047 29.6 + 143.89 Here the "diff. for 1-" increases 0'M89 in 1*; the half of the interval for which the correction is to be computed is 14" 48* = EPHEMERIS. 75 0*.25; we therefore find the value of the first difference at 12* 14" 48% by adding to its value taken for 12* the quantity 0'M89 X 0.25, and then proceed as in simple interpolation. This exam- ple suffices to illustrate the method in all cases where the first difference is given in the Ephemeris for the time against which it stands. In using the British Nautical Almanac and other works of the same kind, interpolation by second differences may be performed by the general interpolation formula here- after given. 64. To find the Greenwich time corresponding to a given right ascen- sion of the moon on a given day. Let T' = the Greenwich time corresponding to the given right ascension a', T = the Greenwich hour preceding T' and corresponding to the right ascension a, Aa = the diff. of E. A. in 1* at the time T, then we have, approximately, rpr rp °' — * To correct for second differences, we have now only to find Aytt = diff. of E.A. in 1* for the middle instant of the interval T—T, and then we have, accurately, rpf rp * — * J^a These formulse give T' — T in minutes of time. 65. To find the distance of the moon from a given object at a given Greenwich time. In the American Ephemeris and the British Nautical Alma- nac, the "lunar distances** are given at every 3d hour of Green- wich time, together with the proportional logarithms of the differ- ences between the successive distances. The proportional logarithm of an angle expressed in hours, &c. is the logarithm of the quotient of 3* divided by the angle ; that of an angle expressed in degrees, &c. is the logarithm of the quotient of 3° divided by the angle. Thus, if A is the angle, in hours. 76 EPHEMERIS. 3* P. L. -4 = log- = log3* —log^ A or, if A is in degrees, P. L. ^ = log-=log3^ — log^ A The angle is always supposed to be reduced to seconds ; so that, whether A is in seconds of time or of arc, we have P. L. ^ = log 10800 — log A Tables of such logarithms are given in works on Navigation. If now we wish to interpolate a value of a lunar distance for a time T-\- 1 which falls between the two times of the Ephemeris 7" and T+ 3*, we are to compute the correction for the interval t and apply it to the distance given for the time T; and if we put J =tbe difference of the distances in the Ephemeris^ J' = the difference in the interval f, we shall have, by simple interpolation, J'= JX- 3* or, by logarithms, log J' = log t + log J — log 3* or, supposing J, J', and i all reduced to seconds, log J' = log f — P. L. J (62) Subtracting both members of this from log 10800, we have P. L. J' = P.L. f + P. L. J (63) which is computed by the tables above mentioned. By (62), however, only the common logarithmic table is required. But the first difterences of the lunar distance cannot be assumed as constant when the intervals of time are as great as 3\ If we put P. L. J = § we observe that Q is variable, and the value given in the Ephe- meris is to be regarded as its value at the middle instant of the interval to which it belongs. If then Of = the value of Q for the middle of the interval t, ^Q = the increase of Q in 3* (found from the successive values in the Ephemeris), SPHEMERIS. 77 we have «.=e-(!^').« (64) in which t is in hours and decimal parts. We find then, with regard to second differences, log J' = logt — Q' Example. — ^Pind the distance d of the moon's centre from the star Fomalhaut at the Greenwich time 1856 March 80, 18* 20»* 24*. Here T= 12*, e = 1* 20- 24* = 1*.34 : ^*'^ """ * ^ = 0.28 : and from the Ephemeris : March 30, 12* ((0 36^ 17' 53" Q, .2993 a C, + 0041 J' — 40 28 — jOOn ,2S At 13* 20- 24* 4 = 35 37 25 ^, .2982 + .0011 lege, 3.6834 log J', 3.3852 66. To find the Greenwich time corresponding to a given lunar dis- tance on a given day. We find in the Ephemeris for the given day the two distances between which the given one falls; and if J' = difference be- tween the first of these and the given one, J = difference of the distances in the Ephemeris, we find the interval ^, to be added to the preceding Greenwich time, by simple interpolation, from the formula J' t = 3*XT or logf = log J' + P.L. J = log J'4- C (65) and, with regard to second differences, the true interval, <', by the formula log^ = logJ'+e' (66) where Q' has the value given in the preceding article. But to find C' by (64) we must first find an approximate value of t. To avoid this double computation, it is usual to find t by (65), and to give a correction to reduce it to <' in a small table which is computed as follows. We have fipom (64), (65), and (66) 78 EPHEMEBIS. log f -log t=Q'-Q = -\^^^^^^yQ By the theory of logarithms, we have, J!f being the modulus ofthe common system, log a: = if [(a: — 1) — J (^ — 1)" + &C.] so that logf-logt = log'-=M^^'-\{'-f-') + kc:^ or, neglecting the square and higher powers of the small fraction This, substituted above, gives jlfX3* ^ 2.VX3* ^ by which a table is readily computed giving the value of V — i [or the correction of / found by (65)], li^'ith the arguments aQ and L Li this formula t and V — t are supposed to be expressed in hours; and to obtain t' — i in seconds we must multiply the second member by 3600 ; this will be effected if we multiply each of the factors t and 3* — < by 60, that is, reduce them each to minutes, so that if we substitute the value of M^= .434294 the formula becomes ^., = «^^-ZlO^(? (67) 2.60676 ^ ^ ^ in which t is expressed in minutes, and V — tin seconds. Example. — 1856 March 30, the distance of the moon and Fomalhaut is 35° 37' 25" ; what is the Greenwich time ? March 30, 12* 0* 0- ((/)=36° 17' 53" Q= .2993 A§= + 41 f= 1 20 3 6 d =35 37 25 log J' = 3.3852 Ap. Gr. time =13 20 36 J' 40 28 \ogt =3.6845 By (07)*,f'— f = —12 True Gr. time = 13 20 24 * Or from the ** Table showing (he correction required on account of the seeond differences of the moon's motion in finding the Greenwich time corresponding to % corrected lunar distance/' which is giyen in the American Ephemeris, and is also included in the Tables for Correcting Lunar Distances giTen in Vol. 11. of thla work. INTERPOLATION IN GENERAL. 79 INTERPOLATION BY DIFFERENCES OF ANY ORDER, 67. When the exact value of any quantity is required from the Ephemeris, recourse must be had to the general interpolation formulae which are demonstrated in analjiiical works. These enable us to determine intermediate values of a function from tabulated values corresponding to equidistant values of the variable on which they depend. In the Ephemeris the data are in most cases to be regarded as functions of the time considered as the variable or argument. Let T^ T-\' Wj 7^+ 2x0^ 7^+ 3t(7, &c., express equidistant values of the variable ; F^ F\ F'\ i^'", &c., corresponding values of the given function ; and let the difterences of the first, second, and following orders be formed, as expressed in the foUo^nng table : — Argument. Function. let Diff. 2d Diff. 3d Diff. 4th Diff. 6t]i Diff. 6th Diff. T F a T-\- w F' a! b C T+2u7 F" a" b' <f d e T+3M7 jpim a'" b" c" d' e / T + 4w7 J^r a»^ b'" d" d" T+6m? F' /iT 6" r+6M7 jrri a The differences are to be found by subtracting downwards^ that is, each number is subtracted from the number below it, and the proper algebraic sign must be prefixed. The difterences of any order are formed from those of the preceding order in the same manner as the first differences are formed from the given func- tions. The even differences (2d, 4th, &c.) fall in the same lines with the argument and function ; the odd differences (1st, 3d, &c.) between the lines. Now, denoting the value of the function corresponding to a value of the argument T+ mo by F^''\ we have, from algebra, jr(»)=JP+na+^ (^-^^ H^ ^^~^^ ^""^^ c+^ ^^-^^ ^^-^^ ^^-^^ d+kc, (68) 1.2 1.2.3 1.2.3.4 in which the coefficients are those of the n** power of a binomial. 80 INTERPOLATION IN GENERAL. In this formula the interpolation sets out from the first of the given functions, and the differences used are the first of their respective orders. If n be taken successively equal to 0, 1, 2, 3, &c., we shall obtain the functions F^ F% -P", F^'% &c., and in- termediate values are found by using fractional values of n. We usually apply the formula only to interpolating between the function from which we set out and the next following one, in which case n is less than unity. To find the proper value of n in each case, let T+ i denote the value of the argument for which we wish to interpolate a value of the function : then nw t t n = — w that is, n is the value of t reduced to a fraction of the interval w. Example. — Suppose the moon's right ascension had been given in the Ephemeris for every twelfth hour as follows : l«t. Diff. D *H R. A. 1856 Blarch 6, 0* 21* 58" 28».89 5, 12 22 27 15.43 6, 22 65 25.50 6, 12 23 23 8.89 7, 23 50 15.63 7, 12 17 9.83 2d DHL 3d Diff. 4th Diff. 1 — 86».97 H- 4'. 79 82.18 6.53 .+ 1'.74 25.65 7.61 1.08 18.04 -f 28* 47'.04 28 10.07 32.18 ' -I-K74 27 37.89 6.53 — 0».66 27 12.24 26 54.20 Required the moon's right ascension for March 5, 6*. 6* 1 Here 7"= March 5, 0*. / = 6\ w = 12\ n = --— = -; and if we denote the coeflicients of a, 6, c, rf, e in (68) by -4, -S, C, -D, U, we have J^ = 21* 58- 28'.39 a = + 28- 47'.04, A=n = J, ^a = + 14 23.52 36.97, 5=^.^^ = — J, Bb = + b=z — c = + 4.79, C=^.^^— ?== + tV Cc= + d = + 1.74, D=C. n = -T^B. ■»'* = - 0.66, E=J).^^—^ = + ,ig,Ee= — 4.62 0.80 0.07 0.02 J'» R A. 1856 March 5, 6» J^'^' = 22 12 66 .74 INTERPOLATION. 81 which agrees precisely with the value given in the American Ephemeris. 68. The formula (68) may also be written as follows : -='+"('+"-fi('+=r(-+"-^'+Tl d -| / e + &c. 2 \ 8 \ 4 \ 6 Thus, in the preceding example, we should have (68*) n — 4 5 n— 3 4 3 n — 1 7 - /u X - 0'.66 = - I (+ 1-.74 + 0-.46) = - ^ (+ 4'.79 — 1'.38) = - 1.38 — 1.71 n = ay — I (— 36..97 — 1'.71) = + 9 .67 i (+ 28«47*.04 + 9*.67) = + 14* 28'.35 and adding this last quantity, 14"» 28'.35, to 21* 58*» 28'.39, we obtain the same value as before, or 22* 12"* 66*. 74. 69. A more convenient formula, for most purposes, may be deduced from (68), if we use not only values of the functions following that from which we set out, but also preceding values ; that is, also values corresponding to the arguments T — Wj T — 2m?, &c. We then form a table according to the following schedule : Tgument. Function. IstDiff. 2dDiff. SdDiff. 4thl>iff. SthDiff. 6th Diff. T 3t£? F T--2W Fu Ki K T— w F. «« «/ h ^. «, T F • a' b d / r+ w F' a" V c" d' T+2w F" a'" b" T+Sw F"> Vol. I.— 6 82 INTERPOLATION. According to the formula (68), if we set out from the function F^ we employ the differences denoted in this table by a', 6', c", &c., and hence for the argument T + nwvfQ find the value of F^*"^ by the formula jrc)^ir+na-+^L(!!z:l) y+ ^ ^"^^^ ^^^) n (n-1) (n^2) (n-3) ^ ^^^ ^ ^ 1.2 1.2.8 1.2.8.4 But we have d!'= d' ^e* = d +€f +€"+/' = d + 2ef+f &c. &c. in which b\ c", &c. are expressed in terms of the differences that lie on each side of a horizontal line drawn in the table immediately under the function from which we set out. These values substituted in the formula give ^ ^ 1.2 1.2.3 (n + l)(n)(n-l)(n-2)^ ^^ (69) ^ 1.2.3.4 ^ ^ in which the law of the coefficients is that one new factor is introduced into the numerator alternately after and before the other factors, observing always that the factors decrease by unity from left to right. The new factor in the denominator, as in the original formula (G8), denotes the order of difference. The inteq>olation by this formula is rendered somewhat more accurate by using, instead of the last difference, the mean of the two values that lie nearest the horizontal line drawn under the middle function : thus, if we stop at the fourth difference, we use a mean between d and r/' instead of d. We thus take into account a part of the term involving the fifth difterenee. Example. — Find the moon's right ascension for 185G March 5, 6*, cnii»l()ying the values given in the Ephemeris for every twelfth hour. This is the same as the example un<ler Art. 67, where it is worked by the primitive fonnula (<)8). But here we take from the Ei>hemeris three values preceding \\\i\i for March 5, 0*, and three values /oWo^nW/ it, and form our table as follows: INTERPOLATION. 83 1866 March 8, 12* 4, l< <l 14 (( <( (t 4,12 6, D'gR. A. 20* 28"» 17'.88 20 58 57.08 21 29 2.01 21 58 28.39 l8t DIff. 2d Diff. 3d Diff. 4th Diff. -1- 80* 89'.20 — 34'.27 80 4.93 — 4'.28 88.55 + 3'. 49 29 26.88 — 0.79 89.34 8.16 5th Diff. — O-.SS 5,12 6, 6, 12 22 27 15.43 22 55 25.50 23 28 8.89 28 47.04 4-2.37 86.97 2.42 28 10.07 82.18 + 4.79 27 87.89 — 0.74 Drawing a horizontal line under the function from which we Bet out, the differences required in the formula (69) stand next to this line, alternately below and above it. a' = + 28- 47'.04, b = — </ = + d = + e = — 39 .34, A = B = A n n 2.37, C = B 3.16, D=C 0.74, E=D 2 n + 1 3 n— 2 4 n + 2 F — 21» 58- 28'.39 h Aa' = + 14 23 .52 h £b =+ 4.92 t'«. CV — 0.15 = + lis. I>d= + = + r* ^HEf E^ = — 0.07 0.01 D's E. A. 1856 March 6, 6* = jP«> = 22 12 56 .74 69*. K in (69) we substitute the values a' = a^-\-b (/ = c, + d &c. we find J.- = F+na,+ (i^±i^6 +(» + D W (n - 1) ^ '^ 1.2 1.2.3 ' (n + 2) (n + 1) (n) (n - 1) + 1727374 ^ + ^'- (TO) in which the law of the coefficients is that one new factor is introduced into the numerator alternately before (aid after the other factors, obsen'ing still that the factors decrease by unity from left to right. The differences employed are those which lie on each side of the horizontal line drawn immediately above the function from which we set out. 84 INTERPOLATION. If in the preceding formulae we employ a negative value of n less than unity, we shall obtain a value of the function between F and F^y and in that case (70) is more convergent than (69). In general, if we set out from that function which is nearest to the required one, we shall always have values of n numerically less than J, and we should prefer (69) for values of n between and + J, and (70) for values of n between and — J, 70. If we take the mean of the two formulse (69) and (70), and denote the means of the odd difterences that lie above and below the horizontal lines of the table, by letters without ac- cents, that is, if we put a = }(«, + a'), c=\{c^^€f) &c. we have F...=i.+„a+^ b+ (!Hd)00(-zl),^ («±1X"!)(ILZL) d + &c. (71) ^ ^2 2.3 2.3.4 ^ ^ ^ The quantities a, c, &c. may be inserted in the table, and will thus complete the row of difterences standing in the same line with the function from which we set out. The law of the coefticients in (71) is that the coefticient of any odd difterence is obtained from that of tlie preceding odd dif- ference by introducing two factors, one at the beginning and the other at the end of the line of factors, observing as before that these factors are respectively greater and less by unity than those next to which they are placed; and the coefticients of the even difterences are obtained from the next preceding even difterences in the same manner. The factors in the denominator follow the same law as in the other formuljs. Example. — Find the moon's right ascension for 1856 March 5, 6\ from the values given in the Ephcmeris for noon and mid- night The table will be as below: INTERPOLATION. 85 Mar. 3, 12» " 4, « 4, 12 " 6, " 5, 12 " 6, " 6, 12 D'8 R. A. 20*28- 17-.88 20 58 57 .08 21 29 2 .01 21 58 28.39 22 27 15 .43 22 55 25 .50 23 23 3 .39 1st Diff. 2(1 Diff. 8d Diff. 4th Diff. + 30«29'.20 — 34-.27 30 4 .93 4-.28 38.55 +3'.49 29 26.38 —0.79 [+29 6 .71] 39.34 [+0 .79] + 3.16 28 47.04 + 2.37 36.97 2.42 28 10.07 32.18 + 4.79 27 37.89 5th Diff. — 0'.33 [— .54] —0.74 Drawing two lines, one above and the other below the func- tion from wliich we set out, and then tilling the blanks by the means of the odd differences above and below these lines (which means are here inserted in brackets), we have presented in the same line all the differences required in the formula (71) ; and we then have b = — c= + d = + F— 21»58"28'.39 » 6'.71, A— n J, Aa— + 14 33 .36 39 .34, 5 — ^' ' 2 + h Bh— 4 .92 n«— 1 0.79, C= A.—^=-ij,, 3.16,2)= 5."^ = --,^^, Cc= — Dd = e=— 0.54, JE?=a. n" 20 = + 2!ff, Ee=^ 0.05 0.02 0.01 jp^oi) = 22 12 56 .75 agreeing within O'.Ol with the value found in the preceding article. Hansen has given a table for facilitating the use of this formula. (See his Tables de la Lune). 71. Another form, considered by Bessel as more accurate than any of the preceding, is found by employing the odd differences that fall next below the horizontal line drawn below the function from which we set out, and the means of the even differences that fall next above and next below this line. Thus, if we put b,= i(b + V), d, = \{d-\- d'), Ac. 86 INTERPOLATION. and combine these with the expressions i<f = i(^l/ ^b), i e' = 1( d'— rf), &c. we deduce b = 1^ — i (f, d = d^ — } e', &c. -l)(n-2) which substituted in (69) give ^ ^ 1.2 "^ 1.2.3 ^ 1.2.3.4 . (n+l)n(n-l)(n~2)(n-}) ^. , + 1.2.3.4.5. +*''• d. (72) To facilitate the application of this formula, draw a horizontal line under the function from which the interpolation sets out, and another over the next following function ; these lines will embrace the odd differences a', c', &c. If we then insert in the blank spaces between these lines the means of the even differ- ences that fall above and below them, we shall have presented in a row all the diftercnces to be employed in the formula. Example. — ^Find the right ascension of the moon's second limb at the instant of its transit over the meridian whose longi- tude is 4* 42"* 19* west from Greenwich, on May 15, 1851. The right ascensions of the moon's bright limb at tlie instant of its upper and lower transits over the Greenwich meridian, are given in the Ephemeris, under the head of '* Moon Culminations." The argument in this case is the longitude, and the intervals of the ar*rument are 12*. The value for any meridian is therefore to be obtained by interpolation, taking for n the quotient obtained by dividing the given longitude (in hours) by 12*. AVe take from the British Nautical Almanac the following values : R.A.^*ii2clHinb. 16* 12- 31>'.a4 15 41 3.41 IG 9 30.8<» lit THir. 2dDiff. SclDiff. 4th Diff. 6th Diir. May 14. U. C. •* 1.'. L. C. " l',. u. c. J 28'" 1 28 24'.37 36.48 -f 12«.ll -f 9.49 — 2'.62 — K58 10 38 !!.•>. 86 17 7 17.12 17 86 8.22 28 45 .97 -f- 6.29 — 0.16 - 4 .20 (-1.42J -f 0».33 " l(i, L. C. »* 10. u. c. - 17. L. C. 28 28 61 .20 61.10 — 6.46 — 1.26 INTERPOLATION. 87 For interpolation by formula (72) we draw a horizontal line below the function from which we set out, and one above the next following function. These lines enclose the odd difterences regularly occurring in the table. Inserting in the blanks in the columns of even diiFerences the means of the numbers above and below, all the differences to be employed in the formula stand in the same line, namely : a' = + 1725'.97, b^= + 7'.B9, c' = — 4'.20, d^ = — l'A2, e' = + 0*.33 As n is here not a simple fraction, the computation will be most conveniently performed by logarithms, as follows : 4» 42- 19* — 16939* log 4.2288878 12* =43200 log 4.6354837 log A — \ogn — 9.5934041 n 0.3921065 9.59340 9.5934 1 9.5934 9.5934 n — 1 — 0.60789 n9.78383 n9.7838 n9.7838 n9.7838 n-J- — 0.10789 n9.0330 n9.0330 n — 2 — 1.6079 n0.2063 wO.2063 n + l = + 1.3921 0.1437 0.1437 9.5934041 (^) 9.69^97 (5)n9.07620 (J) 9.2218 (^'5) 8.6198 (^1^)7.9208 (^) (C) 7.6320 (D) 8.3470 (jE:)n6.6810 («') 3.2370332 2.8304373 (6o) 0.86864 n9.94484 (c')n0.6232 n8.2552 (Jo)n0.1523 (e') 9.5185 n8.4993 n6.1995 Aa' — 11- 16-.764 Bb^ — .879 Of =— 0.018 Dd^ = .032 E^ = 0.000 Increase of E. A. — 11 15.835 E. A. Greenwich Cn E. A. on given merit ilm. — 16» 9- 39-.890 iian — 16* 20- 55'.725 The use of Bessel's formula of interpolation is facilitated by a table in which the values of the coefficients above denoted by Ay jB, Oy D, &c., and also their logaritlmis, are given with the argument 71. 72. Interpolation into the middle. — ^When a value of the function is sought corresponding to a value of the argument which is a 88 ISTERPOLATION. mean between two values for which the function is ^ven, that is, when » = |, we have by (72), since n — | = 0, or, since F+ia' = i {F+ F'), F^ = HF+ F') - i [6o - ^% [i„ - 2\ (/o - &c.)]] (73) which is known as the formula for interpolating into the middle. When the third differences are constant, rf^,, /q, &c. are zero, and the rule for interpolating into the middle between two func- tions is simply : From the mean of the two functions subtract one- eighth the mean of the second differences lohich stand against the func- tions. Interpolation by this rule is correct to third differences inclusive. The formula (73) is especially convenient in computing tables. Values of the function to be tabulated are directly computed for values of the argument difiering by 2'"u?; then interpolating a value into the middle between each two of these, the arguments now difier by 2"*~*m? ; again interpolating into the middle between each two of the resulting series, we obtain a series with argu- ments differing by 2"*"*«/? ; and so on, until the interval of the argument is reduced to 2'*~"'ir or w. Example. — Find the moon's right ascension for 1856 March 5, 6*, from the values of the Ephemeris for noon and midnight. This is the same as the example of Art. 69 ; but, as 6* is the middle instant between noon and midnight, the result will be obtained by the formula (73) in the following simple manner. We have from the table in Art. 69 b^ = - 38'.16 ^(J^+ r) = 22» 12« 51'.91 (/„ = + 2'.79, — -j»5 <^o = — ^J^ 38.68 X i = +4.83 — 38.08 F'^^=22 12 56.74 73. In case we have to interpolate between the last two values of a given series, we may consider the series in inverse order, tlie arguments being T, T—w, T—2w, &c., 7* being the last argument. The signs of the odd differences will then be changed, and, taking the last differences in the several columns as a, 6, c, d^ &€., the interpolation will be effected by (68). IXTERPOLATION. 89 74. The interpolation formvlos arranged acco^'ding to the powers of the fractional part of the argument. When several values of the fuuetion are to be inserted between two of the given series, it is often convenient to employ the formula arranged according to the powers of n. Performing the multiplications of the factors indicated in (68), and arranging the terms, we obtain l?^*> = i?'+n(a — i^ + ^c — id + J« — Ac.) + j^ (6 -c + H (f - I 6 + &c.) -4 (e — kc") ^1.2.3.4.5^ ^ + &C (74) where the differences are obtained according to the schedule in Art. 67. Transforming (71) in the same manner, we have 2?^-) = 7?^+ n (a - J c + 5I5 e — &c.) H (c — i e + &c.) ^1.2.3^ ^ ^ H (d — &c.) ^1.2.3.4^ ^ H (e — &c.) ^1.2.3.4.5^ ^ + &e (75) where the differences a, c, e, are the mean interpolated odd dif- ferences in the line of the function F of the schedule Art. 69. 75. Derivatives of a tabulated function. — ^When the analytical ex- pression of a function is given, its derivatives may be directly found by successive differentiation ; but when this expression is not known, or when it is very complicated, we may obtain values of the derivatives, for particular values of the variable, from the tabulated values of the functions by means of their differences. Denoting the argument hy T + nWy its corresponding function 90 INTERPOLATION. by / ( 2' + w?(?), the successive derivatives of this function cor- responding to the same value of the argument will be denoted by f\T+nw\ f'\T+mo\ f'\T+ nw\ &c., and /(T), f\T)^ f'\T\ &c., will denote the values of the function and its derivatives corresponding to the argument T, or when n = 0. Hence, if we regard nw as the variable, we shall have, by Mao- laurin's Theorem, /(T + nw) = f{T) +fXT) nw +/"(r)^ + &c. Comparing the coefficients of the several powers of n in this formula with those in (74), we have /'(T) = — (a — } 6 + i c — M + i « — &c.) to f'(T)=^(d — 2e + &c.) /'(r)=-^(e-&o.) &c. &c (76) the differences being taken as in Art. 67. Still more convenient expressions are found by comparing Maclaurin's Theorem with (76) ; namely : fXT)=~ (a-Jo+,'o6-&c.) w /"(r)=-l(6-yjd + &c.) /"'(r)=~ (c_tc + &c.) ur r(r)=-]|^(e-&c.) &c. &c. (77) the differenecH being found according to the schedule in Art 69, and the odd differences, a, c, e, &c., being interpolated means. STAR CATALOGUES. 91 The preceding formulae determine the derivatives for the value T of the argument. To find them for any other value, we have, by differentiating Maclaurin's Formula with reference to nw, /'(T + nw) =f\T) +f'iT) . nw + if'^T) . nW + &c. (78) in which we may substitute the values o{ f{T)jf'{T)y &c. from (76) or (77). In like manner, by successive diflerentiations of (78) we ob- tain /" iT+nw) =/" (T) +/'" (T). nw + if^ (T). n»M?' + &c. /'" (T + nw) ==/'" (T) +f' (T). nw + &c. &c. &c. 76. An immediate application of (76) or (77) is the compu- tation of the differences in a unit of time of the functions in the Ephemeris ; for this difterence is nothing more than the first derivative, denoted above by the symbol /'. Example. — Find the difference of the moon's right ascension in one minute for 1856 March 5, 0*. We have in Art. 70, for T = March 5, 0*, a = 29" 6'.71, (? = + O-jg, € = — 0'.54, and m; = 12* = 720^ Hence, by the first equation of (77), f(T) = ^iji (29* 6'.71 — 0-.13 — 0'.02) = 2'.4258 On interpolation, consult also Encke in the Jahrbuch for 1830 and 1837. STAR CATALOGUES. 77. The Nautical Almanac gives the position of only a small number of stars. The positions of others are to be found in the Catalogues of stars. These are lists of stars arranged in the order of their right ascensions, with the data from which their apparent right ascensions and declinations may be ob- tained for any given date. The right ascension and declination of the so-called fixed stars are, in fact, ever changing: 1st, by precession, nutation, and aberration (hereafter to be specially treated of), which are not changes in the absolute position of the stars, but are either changes in the circles to which the stars are referred by sphe- rical co-ordinates (precession and nutation), or apparent changes arising from the observer's motion (aberration); 2d, by the 92 STAR CATALOGUES. proper motion of the stars themselves, which is a real change of the star's absolute position. In the catalogues, the stars are referred to a mean equator and a mean equinox at some assumed epoch. Tlie place of a star so referred at any time is called its mean place at that time ; that of a star referred to the true equator and true equinox, its true place ; that in which the star appears to the observer in motion, its apparent place. The mean place at any time will be found from that of the catalogue simply by applying the preces- sion and the proper motion for the interval of time from the epoch of the catalogue. The true place \\\\\ then be found by correcting the mean place for nutation ; and finally the appa- rent place will be found by correcting the true place for aber- ration. To facilitate the application of these corrections, Bessbl pro- posed the following very simple arrangement. He showed that if a^,, d^ = the star's mean right asc. and dec. at the beginning of the year, a,d= the apparent right asc. and dec. at a time t of that year, T = the time from the beginning of the year expressed in decimal parts of a year, fi^ y! z=i the annual proper motion of the star in right asc. and dec. respectively, then, a z= a^^ Tfi + Aa + Bh -^ Cc + Bd -^ E \ d = d^+rii'+Aa' + Bb' + Cd + Dd! ] (™) in which a, 6, c, rf, a', 6', c', d' are functions of the star's right ascension and declination, and may, therefore, be computed for each star and given \\\\\i it in the catalogue ; -4, By C, /), E are functions of the sun's longitude, the moon's longitude, the longitude of the moon's ascending node, and the obliquity of the ecliptic, all of which depend on the time, so that -4, -B, C, D, E may be regarded 8imi)ly as functions of the time, and given in the Nautical Almanac for the given year and day; £ is a very small correction, usually neglected, as it can never ex- ceed 0".05. If the catalogue does not give the constants a, 6, c^ dy a', ft', c', d'y they may be computed, for the year 1850, by the following formulae (see Chap. XL p. 648): STAR CATALOGUES. 98 a = 46".077 + 20".056 sin a tan d a' = 20".056 cos a b = cos a tan d }/ = — sin a c = cos a sec a d = tan c cos ^ ~ sin a sin d rf =: sin a sec d d! = cos a sin d in which e = obliquity of the ecliptic. Or we may resort to what are usually called the independent constantSy and dispense with the «, i, e, rf, a', 6', (?', d' altogether, proceeding then by the formula « = ««+ ^M +/ +^ sin((y + a)tan^ + ^sin(^+a)8ec^l J == ^0 + '/*' +^ ^^8 ^ + ^ cos (G -j- a) 4" ^ cos (^+ a) sin ^ J ^ ^ the independent constants /, g^ G, A, If, i being given in the Epheraeris, together with the value of r for the given date, expressed decimally. It should be observed that the constants a, 6, c, rf, a', 6', c', d' are not absolutely constant, since they depend on the right ascension and declination, which are slowly changing : unless, therefore, the catalogue which contains them gives also their variations, or unless the time to which we wish to reduce is not very remote from the epoch of the catalogue, it may be prefer- able to use the independent constants. In forming the products Aa^ Bby &c., attention must of course be paid to the algebraic signs of the factors. The signs of J., i5, C, D are, in the Ephemerides, prefixed to their logarithms ; and the signs of a, 6, c, &c. are in some catalogues (as that of the British Association) also prefixed to their logarithms; but I shall here, as elsewhere in this work, mark only the logarithms of negative factors, prefixing to them the letter n. It should be remarked, also, that the B. A. C* gives the * B. A. C. — British Association Catalogue^ containing 8377 stars, distributed in aU parts of the heavens ; a very usefUl work, but not of the highest degree of precision. The Greenwich Catalogues, published from time to time, are more reliable, though less comprehensive. For the places of certain fundamental stars, see Bessel's TabuUe Regiomontanx and its continuation by Wolfers and Zech. Lalande*s Jlittoire Cileste contains nearly 50,000 stars, most of which are em- braced in a catalogue published by the British Association, reduced, under the direction of F. Baily, from the original work of Lalande. The Konigsberg Observa- tions embrace the series known as Bessel's Zones, the most extensive series of observations of small stars yet published. The original observations are given with dat« for their reduction, but an important part of them is given in Weis9b*s Posi- tiones Biediae Stellarum fixarum in Zonis Regiomontanis a Besselio inter — 15*^ et -{-Ib^ deelin. ohservat., containing nearly 32,000 stars. See also Struve*s Catal. generalis, and the catalogues of Aboelandeb, Rumker, 04 STAR CATALOGUES. north polar distance instead of the declination, or r^= 90*^ — 8^\ and, since ;r decreases when d increases, the corrections change their sign. This has been provided for by changing the signs of /i', rt', 6', c'j d' in the catalogue itself. Moreover, in this cata- logue, «, 6, a', b' denote Bessel's c, rf, c', d\ and tice versa ; and to correspond with this, the -4, B^ C, D of the British Almanac denote Bessel's C, -D, -4, ^. The same inversion also exists in the American Ephemeris prior to the year 1865, but in the volume for 1865 the original notation is restored. Example. — Find the apparent right ascension and declination of a Tauri for June 15, 1865, from Argelander's Catalogue. This star is Argel. 108 ; whence we take for Jan. 1, 1830. Mean R. A. = 4^ 2G« 10'.43 Ann. prcc. = + S-. 428 1 ^^^ 35 ^^ Prop, motion =: -|- . 005 i = 4-2 0.15 5 Jan. 1, 1805,0^=4 28 10.585 Mean DecL = -f IS* y SS^.O — 0.17/ "^ 4 30.55 «t, = 4- 16 14 6 .55 We next take the logarithms from the Catal. logs, a 0.5352 from Am. Ephcm. ^ j^^ ^ ^ -g.. for June 15, ISGo, > from the Catal. logs, a' 0.8034 h 7.8704 B 0.0437 b- n0.0607 c 8.4320 CnO.2125 f* 0.2010 d 8.S058 D fil.3089 ^ 9.0378 logs. Aa 0.3220 lib 38231 Cc n8.r,4.34 Dd »i0.1147 logs. Aa' 0.C811 Bb' n0.9O44 Cc fiO.4144 /;<f ttO.3467 to Corr. of a^ yla = -i- 2M03. /?A = -f (:'.007, Cc ^ - 0».044, />J = — l'.S02 Corr. of <V ^«' = -r •* "-80, i?6' = — 8 ".02, CV = — 0".2G, Dd ^ ^ 2".22 We have also fi-om the catalogue /i ~ + 0*.005, //' — — O'MT. The fraction of a year for June 15, 18G5, is r — 0.4G ; and hence Jan. 1, 1865. a^ = 4* 28- 10».585 Sum of corr. of a^ = + 0.824 r/i = -L .(K)2 June 15, 18»;5 a = 4~28 11 .411 \ r^ -f 16° 14* 6".55 Sum of corr. of t\ -- 9 U -= - 5 .70 — .08 16 14 .77 78. ^\lien the greatest precision is required, we should con- sider the change in the Htar s i»laoe even in a fraction of a day, and therefore aLso the change while the star is pa.«sing from ono meridian to another; also the set-ular variation and the changes PiAZZi. Santixi ; and the published ohitervationii of the principal obserratories. 8m also a list of catalogues in the introduction to the B. A. C. THE EARTH. 95 in the precession and in the logarithms of the constants. Fur- ther, it is to be observed that the annual precession of the cata- logues is for a mean year of 865"^ 5*.8. But for a fuller consider- ation of this subject see Chapter XI. CHAPTER ni. FIGURE AND DIMENSIONS OP THE EARTH. 79. The apparent positions of those heavenly bodies which are within measurable distances from the earth are diiFerent for ob- servers on different pai-ts of the earth's surface, and, therefore, before we can compare observations taken in diiFerent places we must have some knowledge of the form and dimensions of the earth. I must refer the reader to geodctical works for the methods by which the exact dimensions of the earth have been obtained, and shall here assume such of the results as I shall have occasion hereafter to apply. The figure of the earth is very nearly that of an oblate spheroid^ that is, an ellipsoid generated by the revolution of an ellipse about its minor axis. The section made by a plane through the earth's axis is nearly an ellipse, of which the major axis is the equatorial and the minor axis the polar diameter of the earth. Accurate geodctical measurements have shown that there are small deviations from the regular ellipsoid ; but it is sufficient for the puq^oses of astronomy to assume all the meridians to be ellipses with the mean dimensions deduced from all the measures made in various parts of the earth. 80. Let EPQP\ Fig. 11, be one of the elliptical meridians of the earth, EQ the diameter of the equator, PP' the polar diameter, or axis of the earth, C the centre, F a focus of the ellipse. Let a = the semi-major axis, or equatorial radius, = CE, b = the semi-minor axis, or polar radius, = CP^ c = the compression of the earth, e = the eccentricity of the meridian. EEDrCTIOX OF LATirUDE. Br the romprcssitm is meant the difference of the equatorial and p»olar radii expressed in parts 7i^ iL of the equatorial radius as unity, or a — 6 , b a a The (cccntrintjf of the meridian is r the distance of either focus from the centre, also expressed in parts of the equatorial radius, or, in Fig. 11, e ^ CF CE But, since PF= CE, we have. tbat is, or CF* PP — PC^ PC* Ue- CE V CE- e» = l — - = 1 — (1— f)» e = I ' 2c — c* (81) Bv a combination of all the most reliable measures, Bessel de^luced the most probable form of the spheroid, or that which luo-t nearly represents all the obse^^•ations that have been made in different parts of the world. He found* a ~ ^ "" 299.1528 or c = wh^mce, by ^81), 2l>9.1528 t = .0816967 lo^' /> = 8.912205 log I (1 — ft ^ = 9.9985458 • .-If/fr/ii'/nii^^A* Xarftnehf^n. No. 4.V. See «1j«o Encke's Tables of the dimensioBi of the terreitrial •pheroi<l id the Jnhrbuck for l8o2. REDUCTION OP LATITUDE. 97 The absolute lengths of the semi-axes, according to Bessel, are, a = 6377397.15 metres = 6974532.34 yds. = 3962.802 miles h = 6356078.96 *' = 6951218.06 " = 3949.555 »* 81. To find the reduction of the latitude for the compression of the earth. Let j4, Fig. 11, be a point on the surface of the earth; AT the tangent to the meridian at that point ; A 0, perpendicular to A Ty the normal to the earth's surface at A. A plane touching the earth's surface at A is the plane of the horizon at that point (Art. 3), and therefore AOy which is perpendicular to that plane, represents the vertical line of the observer at A. This vertical line does not coincide with the radius, except at the equator and the poles. If we produce CE^ OA^ and GA to meet the celestial sphere in E\ Z^ and Z* respectively, the angle ZO'E' is the declination of the zenith, or (Art. 7) the geographical latitude^ and Zis the geographical zenith ; the angle Z'CE' is the declination of the geocentric zenith Z'^ and is called the geocentric or reduced latitude; and ZAZ' = CAO is called the reduction of the latitude. It is evident that the geocentric is always less than the geogra- phical latitude. Kow, if we take the axes of the ellipse as the axes of co-ordi- nates, the centre being the origin, and denote by x the abscissa, and by y the ordinate of any point of the curve, by a and b the semi-major and semi-minor axes respectively, the equation of the ellipse is = 1 If we put f> = the geographical latitude, fp' == the geocentric ** we have, since f is the angle which the normal makes with the axis of abscissae, dx tan f> = dy and from the triangle A CBy tan f ' = — X Vol. L— 7 98 REDUCTION OF LATITUDE. Differentiating the equation of the ellipse, we have _y__ ^ dx X a' dy or tan ^ = — tan f = (1 — e*) tan ^ (82) which determines the relation between f and f '. To find the difference y — f ', or the reduction of the latitude, we have recourse to the general development in series of an equation of the form tan X =p tan y which [PL Trig. Art 254] is X — y = J sin 2y + t J* sin 4y + &c. in which q =^ ^ p + 1 Applying this to the development of (82), we find, after divid- ing by sin V to reduce the terms of the series to seconds, 9 — 9* = — -At;, Bin 2.9 — ;r^-^, sin ^9 — &c. (88) sm 1 ' 2 sm 1" in which ;> — 1 1 — c» — 1 e ? = p + 1 1— e»+l 2— 6» Employing Bessel's value of r, we find ^ = 690".65 ^ — = — r.l6 sin 1" 2 sin 1" and, the subsequent terms being insensible, ^ — ^' = C00".65 sin 2f> — 1".16 sin 4f> (88*) by which ^ — ^' is readily computed for given values of f>. Its value will be found in our Table III. Vol. 11. for any given value of 9. Example. — Find the reduced latitude when ip = 85®. We find by (83), or Table HI., ^ _ ^ = 648".25 = IC 48".25 and hence the reduced or geocentric latitude 9' = 34° 49' 11".75 RliDItS OF THE EARTH. 99 82. To find the radius of the terrestrial spheroid for a given latitude. Let p = the radius for the latitude ^ = AC. Vfe have To express x and y in terms of f , we have from the equation of the ellipse and its diflferential equation, after substituting 1 — c* for -19 if 3? H ^ = a« i?. = (1 — c«) tan 9> X from which by a simple elimination we find a cos cp and hence l/(l — €* sinV) (1 — c^ a sin^ l/(l — e" sinV) / r 1 — 2 6* siii*f> + e* BJnV ] ''""^ VL l~6«sinV J (84) by which the value of p may be computed. The logarithm of />, putting a = 1, is given in our Table m. Vol. 11. But the logarithm of p may be more conveniently found by a series. If in (84) we substitute e = \ —p sin's? = J (1 — cos 2f) we find, putting a = 1, ^ /[ !+/* + (!-/*) cos 2y> 1 ^Ll+/' + (l-/')C08 2j 1+/* 1+/ + /' + (l-/')C08 2v,. ^+(;-=^F^^H^.j Now (PI. Trig. Art. 260) if we have an expression of the form X=|/(l + m' — 2mcos C) {A) 100 RADIUS OF THE EABTH. we have, if -Jf = the modulus of the common system of loga- rithms, 1 TT umI n i tn* COB 2C , m»cos3C , . \ ,^ log Jr=— -MImcos 0-\ 1 1- &c. I (5> by which we may develop the logarithms of the numerator and denominator of the above radical. Hence we find log p = log + Ml (m — m!) cos 2^ cos 4f -| cos 6^ — &c. j in which we have put for brevity 1-/' , 1-/ m = — m == - 1+/' 1+/ Restoring the value of /= |/(1 — 6*) and computing the numerical values of the coefficients, we find log p = 9.9992747 + 0.0007271 cos 2?) — 0.0000018 cos 4^ (85) as given by Encke in the Jahrbuch for 1852. The values of p and <p' may also be determined under another form which will hereafter be found useful. We have in Fig. 11, p sin (p^ =^y^ p cos ^' = x, or a (\ — e*) sin «p p Bm 0' = — ^^- : »/(l-e'8inV) . (gg) a cos I \ ^ p cos = ^ ^ |/(1 — e» sin' if) which may be put under a simple form by introducing an auxi- liary 4'? so that sin 4 = e sin ^ p sin ^' = a (1 — e*) sin ^ soc 4 \ (87) ^ cos (p' =: a cos ^ soc 4 "We can also deduce from these, p sin (^ — ^*) = I fle* sin 2f sec 4 /> cos (^ — ^) = a cos 4 } (88) NORMAL. 101 Hence, also, the following: , = aJ( r^ ^) (89) \ \ COS ^' COS (^ — /) / ^ "^ 83. To find the length of the normal terminating in the axis^ for a given latitude. Putting N= the normal = AO (Fig. 11), we have evidently N= ^.^^ = — — ? (90) COS y> ^(1 — e* sin' ^) or, employing the auxiliary 4' of the preceding article, N= a sec 4 84. To find the distance from the centre to the intersection of the normal with the axis. Denoting this distance by ai (so that i denotes the distance when a = 1), we have in Fig. 11, ai = CO and, from the triangle ACOy P sin (cp — «>') at = — ^-^ cos <p or, by (88), ae* sin ^ , . ,_,^ ax = -77^ ., . , , = ae^ sm cp sec 4 (91) l/(\ — e* sin* tp) ^ ^ ^ ^ 85. To find the radius of curvature of the terrestrial meridian for a given latitude. — Denoting this radius by if, we have, from the dif- ferential calculus, where we employ the notation D^y^ JD/ y to denote the first and second dilterential coefficients of y relatively to x. We have from the equation of the ellipse -TV h^ X ^ 6* ^xy = --;T.-^ J^:y = a* y * ^ a^rf 102 DEVIATIONS OP THE PLUMB LINE. whence Observing that 6^ = a* (1 — e^), we find, by substituting the values of X and y in terms of ^ (p. 99), ^ = 71^^ (92) (1 — e* sin' ^)t Example. — ^Find the radius of curvature for the latitude of Greenwich, <p = 51° 28' 38".2, taking a = 6377397 metres. We find R = 6373850 metres. 86. Abnormal deviations of the plumb line. — Granting the geo- metrical figure of the earth to be that of an ellipsoid of revolu- tion whose dimensions, taking the mean level of the sea, are as given in Art. 80, it must not be inferred that the direction of the plumb line at any point of the surface always coincides precisely with the nonnal of the ellipsoid. It would do so, indeed, if the earth were an exact ellipsoid composed of perfectly homoge- neous matter, or if, originally homogeneous and plastic, it has assumed its present form solely under the influence of the attraction of gravitation combined with the rotation on its axis. But experience has shown* that the phimb line mostly deviates from the normal to the regular ellipsoid, not only towards the north or south, but also towards the east or west ; so that the apparent zenith as indicated by the plumb line difters from the true zenith corresponding to the normal both in declination and right ascension. These deviations are due to local irregularities both in the figure and the density of the earth. Their amount is, however, very small, seldom reaching more than 3" of arc in any direction. In order to eliminate the influence of these deviations at a given place, observations are made at a number of places as nearly as possible symmetrically situated around it, and, as- suming the dimensions of the general ellipsoid to be as we have given them, the direction of the plumb line at the given place is deduced from its direction at each of the assumed places (by ♦ U.S. Coast Surrey Report for 1863, p. 14» RSDUCTIOK TO THE CENTRE OF THE EARTH. 108 the aid of the geodetic measures of its distance and direction from each) ; or, which is the same thing, the latitude and longi- tude of the place are deduced from those of each of the assumed places : then the mean of all the resulting latitudes is the geodetic htiiude, and the mean of all the resulting longitudes is the geodetic longitude^ of the place. These quantities, then, correspond as nearly as possible to the true normal of the regular ellipsoid ; the geodetic latitude being the angle which this normal makes with the plane of the equator, and the geodetic longitude being the angle which the meridian plane containing this normal makes with the plane of the first meridian. The geodetic lati- tude is identical with the geographical latitude as we have defined it in Art. 81. The astronomical latitude of a place is the declination of the apparent zenith indicated by the actual plumb line ; but, unless when the contrary is stated, it will be hereafter understood to be identical with the geographical or geodetic latitude. It has recently been attempted to show that the earth differs sensibly from an ellipsoid of revolution;* but no deduction of this kind can be safely made until the anomalous deviations of the plumb line above noticed have been eliminated from the discussion. CHAPTER IV. REDIJCnON OP OBSERVATIONS TO THE CENTRE OF THE EARTH. 87. The places of stars given in the Ephemerides are those in which the stars would be seen by an observer at the centre of the earth, and are called ^eoeew^nc, or true, places. Those observed ftt)m the surface of the earth are called observed, or apparent, places. It must be remarked, however, that the geocentric places of the Ephemeris are also called apparent places when it is intended ♦ See Attr. Naeh. No. 1308. 104 PARALLAX. to distinguish them from mean places, a distinction which will be considered hereafter (Chap. XI.). It will also be noticed that we frequently use the terms true and apparent as relative terms only ; as, for example, in treating of the eftect of parallax, the place of a star as seen from the centre of the earth may be called true, and that in which it would be seen from the surface of the earth were there no atmosphere, may in relation to the former be called apparent; but in considering the effect of refraction, the star's place as it would be seen from the surface of the earth were there no atmo- sphere may be called true, and the place as affected by the re- fraction may in relation to the former be called apparent; and similarly in other cases. PARALLAX. 88. The parallax of a star is, in general, the difference of the directions of the straight lines drawn to the star from two different points. The difference of direction of two straight lines being simply the angle contained between them, we may also define parallax as the angle at the star contained by the lines drawn to the two points from which it is supposed to be viewed. In astronomy we frequently use the term parallax to express the difference of altitude or of zenith distance of a star seen from the surface and the centre of the earth respectively; and, in order to express parallax in respect to other co-ordi- nates, proper qualifying terms are added, as " parallax in decli- nation," &c. Assuming (at first) the earth to be a sphere, let -4, Fig. 12, be the position of the observer on its surface, C* the centre, (LIZ the vertical line, and S vl star within a measurable distance OS' from the centre. AJl^ a tangent to the surface at A, and (V/, parallel to it, drawn through the centre, may each be regarded as lying in the plane of the celestial horizon (note, p. lU). The true or geocentric altitude of the star above the celestial horizon is then the angle iSY7/, and the apparent altitude is the angle SAJP. In this case the directions of the star from C and from -1 are compared with each other by referring them to two PARALLAX. 105 lines which have a common direction, i.e. parallel lines. But a still more direct method of comparison is obtained by referring them to one and the same straight line, as CAZ, Z being the zenith. We then call ZCS the true and ZAS the apparent zenith distance, and these are evidently the complements of the tnie and apparent altitudes respectively. In the figure we have at once ZAS—ZCS = ASG that is, the parallax in zenith distance or altitude is the angle at the star subtended by the radius of the earth. When the star is in the horizon, as at H\ the radius, being at right angles to AH\ subtends the greatest possible angle at the star for the same distance, and this maximum angle is called the horizontal parallax. The equatorial horizontal parallax of a star is the maximum angle subtended at the star by the equatorial radius of the earth. 89. To find the equatorial horizontal parallax of a star at a given distance from the centre of the earth. Let r = the equatorial horizontal parallax, J = the given distance of the star from the earth's centre, a = the equatorial radius of the earth, we have from the triangle CAH' in Fig. 12, if CA is the equatorial radius. « (93) sin r = _ J The value of n given in the Ephemeris is always that which is given by this formula when for J we employ the distance of the star at the instant for wliich the parallax is given. 90. To find the parallax in altitude or zenith distance^ the earth being regarded as a sphere. Let Z =z the true zenith distance = ZCS (Fig. 12), C' = the apparent zenith distance = ZAS, p z= the parallax in alt. or z. d. = CSA, 106 PARALLAX. The triangle SAC gives, observing that the angle SAC =; 180° — C', sin /> a - — ^ = -7 = Sin r sin ; J sin j> = sin (C — Z) = sin r sin C' (94) or, If we put A == the trae altitude, A' = the apparent altitude^ then it follows also that sin /) = sin (A — A ') = sin r cos A' (95) Except in the ease of the moon, the parallax is so small that wo may consider ;r and p to be proportional to their sines [PL Trig. Art. 55] ; and then we have /) = r sin r =^ ^ cos A' (96) Since when f ' = 90° we have sin ^' = 1, and when f ' = 0, sin ^' =z 0, it follows that the parallax is a maximum when the star is in the horizon, and zero when the star is in the zenith. Example. — Given the apparent zenith distance of Venus, f' = 6-4° 43', and the horizontal parallax ;r^20".0; find the geocentric zenith distance. log r 1.3010 C = 64^ 43' 0"0 log 8in :' 9.9563_ p= 18.1 log/) 1.2573 C = 04 42 41.9 Wlicn the true zenith distance is given, to compute the paral- lax, we may first use this true zenith distance as the apparent, and tin<l an a|>proximate value of p by the formula p = z sin J; then, taking tlie ai>pr()ximate value of C' -- C ~ /^» ^^'^* compute a more exact value of p by the fonnula (94) or (OG). This second a|)pr<)ximati()n is unnecessary in all cases exce|>t that of the moon, and the parallax of the moon is so great that it becomes necessary to take into account the true fii^ure of the earth, as in the following more general investigation of the subject. 91. Tn consefjuence of the spheroidal figure of the earth, the vertical line of the observer docs not pass through the centre, and therefore the geocentric zenith distance cannot be directly PARALLAX. 107 referred to this line. If, however, we refer it to the radius drawn from the place of observation (or CAZ'y Fig. 11), the zenith dis* tanee is that measured from the geocentric zenith of the place; whereas it is desirable to use the geographical zenith. Hence we shall here consider the geocentric zenith distance to be the angle which the straight line drawn from the centre of the earth to the star makes with the straight line drawn through the centre of the earth parallel to the vertical line of the observer. These two vertical lines are conceived to meet the celestial sphere in the same point, namely, the geographical zenith, which is the common vanishing point of all lines perpendicular to the plane of the horizon. Thus both the true and the apparent zenith distances will be measured upon the celestial sphere from the pole of the horizon. The azimuth of a star is, in general, the angle which a vertical plane passing through the star makes with the plane of the meri- dian. When such a vertical plane is drawn through the centre of the earth, it does not coincide with that drawn at the place of observation, since, by definition (Art. 3), the vertical plane passes through the vertical line, and the vertical lines are not coincident. Hence we shall have to consider a parallax in azimuth as well as in zenith distance. 92. To find the parallax of a star in zenith distance and azimuth tchai the geocentric zenith distance and azimuth are given y and the earth is regarded as a spheroid.*^ Let the star be referred to three co-ordinate planes at right angles to each other : the first, the plane of the horizon of the observer; the second, the plane of the meridian; the third, the plane of the prime vertical. Let the axis of x be the meridian line, or intersection of the plane of the meridian and the plane of the horizon ; the axis of y, the east and west line ; the axis of ^, the vertical line. Let the positive axis of x be towards the south; the positive axis of y, towards the west; the positive axis of Zy towards the zenith. Let J' = the distance of the star from the origin, which is the place of observation, C' = the apparent zenith distance of the star, A' = the apparent azimuth " " " * The inyestigation which follows is nearly the some as that of Olbebs, to whom the method itself is due. 108 PARALLAX. then, 0^ y\ z' denoting the co-ordinates of the star in this system, we have, by (39), of = A* sin C* cos A y' = J' sin C' sin A 2' = J' cos C' Again, let the star be referred by rectangular co-ordinates to another system of planes parallel to the former, the origin now being the centre of the earth. In the celestial sphere these planes still represent the horizon, the meridian, and the prime vertical. K then in this system we put J = the distance of the star from the origin, C = the true zenith distance of the star, A = the true azimuth " " and denote the co-ordinates of the star m this system by x, y, and z, we have, as before, a: = J sin C cos A y = J sin C sin A z =z J cos C Now, the co-ordinates of the place of observation in this last system, being denoted by a, 6, c, we have a =^ /o sin (^ — ^') 6 = c = p cos (^ — f ') in which /> = the earth's radius for the latitude f of the place of observation, and (p' is the geocentric latitude, f — <f' being the reduction of the latitude. Art. 81 ; and the formula? of transforma- tion from this second system to the first arc (Art. 33) X = xf -{- a y =: y' -^ b z = z' -{■ c or, x^ z= X — a y' = y — ^ z' = z — c whence, substituting the above values of the co-ordinates, J' sin r cos A^ = J sin C cos A — /> sin (^ — f ') ^ J' sin :' sin A' =^ J sin : sin A > (07) J' cos C* = J cos C — p cos (^ — s?') J which arc the general relations between the true and apparent zenith distances and azimuths. All the quantities in the second members being given, the first two equations determine J'sin ^', and A '; and then from this value of J'sin ^', and that of J'cos ^' given by the third equation, J' and J' are determined. PARALLAX. . 109 But it is convenient to introduce the horizontal parallax instead of J. For, if we put the equatorial radius of the earth = 1, we have 1 sm TT = — J and hence, if we divide the equations (97) by J, and put /=^ we have / sin ^ cos A' = sin C cos A — /o sin tt sin (jp — <p') \ f sin C' sin A' = sin C sin A v (98) / cos C' = cos C — p Am: cos (^ — /) j To obtain expressions for the difference between ^ and f ' and between A and A'^ that is, for the parallax in zenith distance and azimuth, multiply the first equation of (98) by sin -4, the second by cos A^ and subtract the first product from the second ; again, multiply the first by cos -4, the second by sin -4, and add the products: we find / sin C sin {A' — A) = /> sin r sin (f> — <p') sin A 1 / sin C' cos (A' — A) = sin C — p sin r sin {<p — <p') cos A / ^ ^ Multiplying the first of these by sin J (A' — A), the second by cos \ (A' —A), and adding the products, we find, after dividing the sum by cos \ (A' — A), cos } (A' + A) / sin C' = sin C — p sin tt sin (^ — ^') cos 1 (A' — A) which with the third equation of (98) will determine f '. If we assume x such that ^ / fN cos J (A' + A) ^^ tan p' = tan (sP — /) ry^T -r (100) cos } (A' — A) we have the following equations for determining f ' : / sin C' = sin C — p sin - cos (^ — ^') tan y f cos C' = cos C — /o sin TT cos (^ — ^') which, by the process employed in deducing (99), give X - rt^ ^ • / /N sin (C — r) / sm (r — C) = ^ s^^ ^ cos (jp — /) - ^ ^ } (101) / cos (C' — C) = 1 — /> sin ir cos (sp — 9') cos p' cos (C — f) cos p' (102) 110 . PARALLAX. By multiplying the first of these by sin J (^' -— ^), the second by cos I (^' — f ), and adding the products, we find, after dividing by cos 4 (C' - C), / o sin r cos {if ~ y') cos [} (C + ~ r] cos p' cos 1 (C' — C) or multiplying by J, J' = J — P ^^^ ^^ — ^') ^^^ 1^^ (^' + ^)-~ ^3 (108) cos Y cos J (C* — C) The equations (99) determine rigorously the parallax in azimuth ; then (100) and (102) determine the parallax in zenith distance, and (103) the distance of the star from the observer. The relation between A and J' may be expressed under a more simple form. By multiplying the first of the equations (101) by cos x^ the second by sin x^ the difterence of the products gives A' = J ^ILSL^IZI) (104) sin (C -^r) 93. The preceding formulae may be developed in series. Put p&inn si n (y> — fQ m = ; — sm C then (99) become / sin C sin (A' — ^) = m sin C sin A f sin C cos (A' — A) = sin C (1 — 7n cos A) whence ^ A, jv ^ sin A ,,y*-x tan (A' — A) = (105) 1 — m cos A and therefore [PL Trig. Art. 258], A' — A being in seconds, ., . msinA , m*8in2A m'sinSA , . ,^^. A' — A = H H + &c. (106) sin 1" ^ 2 sin 1" ^ 3 sin 1" ^ ^ ^ To develop j^ in series, we take , , ,^ cos [A + iU'- ^)] tan X = tan (^p — f ') ?^ — -— ^ — ^ cos } (il — A) = tan (v> — ^') [cos A — sin A tan } (A' — A)] whence, by interchanging arcs and tangents according to the PARALLAX. Ill formulse tan~^ ^ = y — J ^ + &c., tan x = x + ^s^ + kc. [PI. Trig. Arts. 209, 218], / #N A (sf> — ^')V sin r sin* ^ sin 1" , ^ .,^_ 2 sin C where the second term of the series is multiplied by sin 1" because y and <p — ^' are supposed to be expressed in seconds. Again, if we put /t) sin rr cos (^ — ^) cos / we find from (102) taD(r-C)= """(g-r) (108) ^ "^ 1 — nco8(C — r) whence, f ' — f being in seconds, nsinC^r) . n'sin2(C~r) . n>8in3(C-~r) . j^^^ ,109) sinl" ^ 2 8inl" ^ 3 sin 1" ^ ^ ^ Adding the squares of the equations (102), we have /* = ( jV = 1 — 2 n cos (C — r) + n« whence [equations {A) and (jB), Art. 82] log J' = log J — Jf (n cos (C — r) + ^* ^^^ ^ (^ ~" ^) . ^. &c.) (110) where -Jf =the modulus of common logarithms. 94. The second term of the series (107) is of wholly inappre- ciable eflect ; so that we may consider as exact the formula ;^ = (^ — /) cos A (111) and the rigorous formulae (105) and (108) may be readily com- puted under the following form : Put . - - p sin r sin (c> — c>') cos A sm^ = m cos -4 = '- ~ ^^ sm C then ^ (112) tan(-4'— -4) = ^5_^-^?iLd == tan * tan (45° + J *) tan il 112 PARALLAX. Put Bin d' = n cos (C — r) = — — ^^^ -^ then . ,w ^. sin 1^' tan (C — r) ) (^^^) tan (r — = ; . '^ 1 — Bin * = tan t>' tan(45*> + } 6')tan (C — r)y Example. — In latitude ^ = 38° 59', given for the moon, A = 320^ 18', ^ -: 29^ 30', and ;r = 58' 37".2, to find the parallax in azimuth and zenith distance. We have (Tahle III.) for f = 38° 59', ip — ip' = 11' 15", log p = 9.999428: hence by (111) ;' = 8'39".3 and f — r = 29° 21' 20".7 ; vdth which we proceed by (112) and (113) as follows : logpsinTT 8.28118 log p sin tt 8.281179 log sin (^ — f ) 7.51488 log cos (^ — f ) 9.999996 log coseo C 0.80766 log sec y 0.000001 log cos A 9.88615 log cos (C — >) 9.940818 ^ = 18", log sin ^ 5.98987 ^'= 51' 1".5, log sin i^ 8.171491 logtani? 5.93987 log tan t^ 8.171589 log tan (450 -f J ,9) 0.00004 log tan (45o 4- } ^) a006446 log tan A n9.91919^ log tan (C — y) 9.760087 log tan {A' — A) n5.8o910 log tan (C — C) 7.928072 A' — A ---. — 14".91 C — C = 29' 7^.79 A = 820° 17' 4ry'.09 ^=290 59' 7^.79 It 18 evident that we may, without a sacrilice of accuracy, omit the factors cos {f — (p') and cos y in the computation of sin &\ If we neglect the compression of the earth in this example, we find by (94) C — (: = 29' 17".9, which is 10" in error. 95. To find the parallax of a star in zenith distance and azimuth when the apparent zniith distance and azimuth are given^ the earth being regardM as a spheroid. If we multiply the fii*st of tlie equations (101) by cos f' and the second by sin ^', the difterence of the products gives a;« rr» r\ _ P ^'" ^ ^Q^ (^ ~ y') ^^'" (^ ~ r^ sin (^, — ^y — COB Y for wliidi, since cos [if — ^') and cos y are each nearly equal to unity, we may take, without sensible error, sin (::' — :)=/> sin TT Bin (C — r) (H*) PARALLAX. 118 in which y has the value found by (111), or, with sujfieient accu- racy, by the formula r = (sp — ?»') cos ^' (115) Again, if we multiply the first of the equations (98) by sin A' and the second by cos A\ the difference of the products gives sin (^' -A-) = PBi""«in(y-/)8in^' sin C to compute which, ^ must first be found by subtracting the value of the parallax ^' — ^, found by (114), from the given value of ^'. Example. — ^In latitude tp = 38° 59', given for the moon A^ = 320° 17' 45".09, C' = 29° 59' 7".79, tt = 58' 37".2, to find the parallax in zenith distance and azimuth. We have, as in the example Art. 94, ^ — ^' = 11' 15", log p = 9.999428, r = {v — v') cos A' = 8' 39".3, C' — r = 29° 50' 28".5 ; and hence, by (114) and (116), log p sin r 8.281179 log p sin r 8.28118 log sin (C — r) 9.696879 log sin (^ — ^') 7.51488 log sin (C — C) 7.928058 log sin A' w9.80588 C — C = 29' 7".79 log cosec C 0.80766 C = 29° 80' 0" log sin {A' — A) n5.85910 A' — A = — 14".91 A = 820° 18' 0" agreeing with the given values of Art. 94. 96. For (lie planets or the sun^ the following formulae are always BuflBiciently precise : c'-c=^;r6in(:'-r) 1 A — A= pit sin (^ — 9?') sin A' cosec C' j v. ^« ; and in most cases we may take C ~Z^^'^ ®^^^ C') ^^^ -^' — ^ = 0. The quantity pr: is frequently called the reduced parallax^ and t: — ^;r = (1 — p):: the reduction of the equatorial parallax for the given latitude ; and a table for this reduction is given in some collections. This reduction is, indeed, sensibly the same as the correction given in our Table XIII., which will be explained more particularly hereafter. Calling the tabular correction a;:, we shall have, with sufficient accuracy for most purposes, pn =^ n — A;r VouL— 8 114 PARALLAX. 97. The preceding methods of computing the parallax enable us to pass directly from the geocentric to the apparent azimuth and zenith distance. There is, however, an indirect method which is sometimes more convenient. This consists in reducing both the geocentric and the apparent co-ordinates to the point m xchich the vertical line of the observer intersects the axis of the earth. I shall briefly designate this point as the point (Fig. 11). We may suppose the point to be assumed as the centre of the celestial sphere and at the same time as the centre of an imaginary terrestrial sphere described with a radius equal to the normal OA (Fig. 11). Since the point is in the vertical line of the observer, the azimuth at this point is the same as the appa- rent azimuth. If, therefore, the geocentric co-ordinates are first reduced to the point 0, we shall then avoid the parallax in azimuth, and the parallax in zenith distance will be found by the simple formula for the earth regarded as a sphere, taking the normal as radius. Since the point is in the axis of the celestial sphere, the straight line drawn from it to the star lies in the plane of the declination circle of the star; the place of the star, therefore, as seen from the point 0, difters from its geocentric place only in declination, and not in right ascension. We have then only to find the reduction of the declination and of the zenith distance to the point 0. Ist. To reduce the declination to the point O. — ^Let PP', Fig. 13, be the earth's axis; C'the centre; the point in which the vertical line or normal of an observer in the given latitude f meets the axis; S the ytar. We have found for CO the expression (Ail. 84) CO = ai in which a is the equatorial radius of the earth, and c* sin ^ Fig. 13. I = |/(1 — c* sin* f) Let J =:^ the star's geocentric distance Jj ~z the star's distance from the point o -.- the geocentric declination ^^ — the declination reduced to the point SC, SO, 90° — PCS, 00° -POS, PARALLAX. 115 then, drawing SB perpendicular to the axis, the right triangles SCB and SOB give J, sin a, == J sin ^ + ai \ J, cos a, = J COB ^ / ^^ •' which determine J^ and 3^. Prom these we deduce Jj sin (^j — ^) = ^i cos 9 Jj cos (^4 — ^) = J -f- flt sin ^ } (119) which determine J^ and the reduction of the declination. If we divide these by J, and put ^ 4i .a f^ = i Bm^ = - in which n denotes, as before, the equatorial horizontal parallax, they become /j sin (^4 — S) =:% sin ^ cos d /j cos (d^ — ^) = 1 -(- t sin TT sin ^ whence ^ , . .. i sin TT cos d tan (a, — ^) = ^ , . . 7—^ 1 -f t sm r sm d or in series [PI. Trig. Art. 257], , ^ t sin TT cos ^ ft sin «)' sin 2 ^ . ., o. — tf = -^ V- &c. sinl" 2sinr' ^ But since the second term of the series involves f* and conse- quently e*, and this is further multiplied by the small factor sin^ ;r, the term is wholly inappreciable even for the moon; and, as the first term cannot exceed 25" in any case, we shall obtain ex- treme accuracy by the simple formula ^1 — ^ = t ff cos a (120) The value of A^ is found from (119), by the same process as was used in finding J' in (103), to be . . f 1 , . . sin J (a, + ^) 1 * \ ^ cosJ(a, — a)/ or, on account of the small difterence between d^ and ^, J, = J (1 + i sin TT sin ^ (121) 116 PARALLAX. As ^1 — ^ is 80 small, it may be accurately computed with logarithms of four decimal places, and it will be conveuient to substitute for i the form in which i = A sin ^ A = |/(1 — e* sin* f ) The value of log A may then be taken from the following table with the argument ^ = the geographical latitude f log^ 0" 7.8244 10 7.8245 20 7.8246 30 7.8248 40 7.8250 50 7.8253 60 7.8255 70 7.8257 80 7.8258 90 7.8259 •We shall then compute 8^ — 3 and Jj under the following forms : 9^ — d = A t: sin ^ cos d J^ = J (1 -\- A sin K sin f> sin d) } (122) K the value of z^ has been found as below, we may take d^ — d = e* r^ sin f cos d 2cL Fig. B A To find the p^irallax in zenith distance for the point 0. — Let ZAOy Fig. 14, be the vertical line of the ob8er\'er at A. The normal ylO terminating in the axis being denoted by iV, we have, by (90), 14. N = a |/(I — e* sin* ^) But if in (84) we write c* sin* <f for c* sin' f , we have p = a |/(1 — e* sin' ^) and this value is sufficiently accurate for the compu- tation of the parallax in all cases. If then we put a — ly we have PARALLAX. 117 P K now in the vertical plane passing through the line ZO and the star S we draw SB perpendicular to OZ, and put Cj = the zenith distance at O = SOZ C' = the apparent zenith dist. = SAZ the triangles 0/SB, ASB give J' COB r = J. COB C- I I ^j23) J' sin C' = ^1 sin d J Dividing these equations by Jp and putting J' 1 — =/ sin TT, == A ' Hi they become /i cos C' = cos Cj — sin ^^ /j sin C' = sin Ci from which we deduce /j sin (C' — Ci) = sin r^ sin C^ /j cos (C' — Cj) = 1 — sin TTj cos Cj tan (C - C.) = , «'""»»'"^« (124) 1 — sm TTj cos Ci and in series, sin ;r, sin C, sin' tt, sin 2 C, ^ ^ ^*- sinl" + 2sinl" ^ ^' ^^^^^ Or, rigorously, sin t* = sin n^ cos C, tan (C — Ct) = tan ^ tan (45^ + J ^) tan C^ To find ^Ti we have 1 sm 7c, = — = or sm TT, = * jo Jj /» J (1 -f- ji sin TT sin ^ sin ^) sin 9r } (126) p(l + A sin TT sin ^ sin S) (127) 118 PARALLAX. But this very precise expression of tt^ will seldom be required: it will generally suffice to take sm ffj = sin^r or TTj P P which will be found to give the correct value of tt^^ even for the moon, within 0".2 in every case. Wliere this degree of accu- racy suffices, we may employ a table containing the correction for reducing n to t:^^ computed by the formula At: =..-«=,(i_i) Table XTEL, Vol. 11., gives this correction with the arguments 7t and the geographical latitude ip. Taking the correction from this table, therefore, we have JTj = TT 4- Ajt (128) Zd, To compute the parallax in zenith distance for the point O when the apparent zenith distance is (jiven. Multiplying the first equation of (123) by sin J^', the second by cos ^', and subtracting, we find 8in(r — C,) = — sinC or sin (C — Cj = sin r^ sin C (129) If we denote the apparent altitude by A' and the altitude reduced to the point hy Aj, this equation becomes sin (Aj — A') = sin r^ cos W (130) Example. — In Latitude ip = 38° 59', given the moon's hour angle t =- 341° 1' 3«".85, geocentric declination d = + W 39' 24".r>4, and the equatorial horizontal parallax ;: — 58' 37".2, to find the apparent zenitli distance and azimuth. The geocentric zenith distance and azininth, computed from tliese data by Art. 14, are Z '- 29° 30', A ■--- 320° 18', whicli are the values employed in our example in Art. 94. To compute PARALLAX. 119 by the method of the present article, we first reduce the declina- tion to the point by (122), as follows : For ^ = 38^ 59' log A 7.8250 IT = 3517".2 log n 3.5462 log sin f 9.7987 ^ = 14*> 39' 24".54 log cos a 9.9856 d^—d= 14 .31 \og(d^—d) 1.1555 a, = 14** 39' 38".85 With this value of 3^ and t = 341° V 36".85, the computation of the zenith distance and azimuth by Art. 14 gives for the point O C, = 29^ 29' 47".67 A, = 320^ 17' 45".09 and this value of J.^ is precisely the same as A' found in Art. 94, as it should be, since the azimuth at the point and at the observer are identical. We find from Table XIH. a;: = 4''.6, and hence ;ri= 58' 37".2 + 4''.6 = 58' 41".8; and then, by (126), log sin TT^ 8.23232 log cos C, 9.93971 d = 51' 5" log sin d 8.17203 log tan »9 8.17208 log tan (45° + i »>) 0.00645 C, = 29*> 29' 47".67 log tan C^ 9.75258 r — C, = 29 20 .03 log tan (C— CJ 7.93111 C' = 29° 59' 7".70 agreeing with the value found in Art. 94 within 0".09. If we had computed t:^ by (127), the agreement would have been exact. 98. To find the parallax of a star in right ascension and declination when its geocentric right ascension and declination are given. The investigation of this problem is similar to that of Art. 92. Let the star be referred by rectangular co-ordinates to three planes passing through the centre of the earth : the first, the plane of the equator ; the second, that of the equinoctial colure ; the third, that of the solstitial colure. Let the axis of x be the straight line drawn through the equinoctial points, positive towards the vernal equinox ; the axis of y, the intersection of 120 PARALLAX. tlie plane of the solstitial eolure and that of the equator, positive towards that point of the equator whose right ascension is 90® ; the axis of z, the axis of the heavens, positive towards the nortL Let a = the star's geocentric right ascension, S = " " declination, J = " " distance, then the co-ordinates of tlie star are X = J cos d cos a y = J cos d Bin a 2 = J sin ^ Again, let the star he referred to another system of planes parallel to the first, the origin being the observ^er. The vanish- ing circles of these planes in the celestial sphere are still the equator, the equinoctial eolure, and the solstitial eolure. Let a' = the star's observed right ascension, if = " " declination, J' = " distance from the observer, where by observed right ascension and declination we now mean the values which diiier from the geocentric values by the paral- lax depending on the position <>f the observer on the surface of the earth. The co-onlinates of the star in this svstem will be X' = J' cos tY cos a i/ =:z J' COS o sin a' y ^ J' sin <r Xow, if 0= the sidereal timer- the right ascension of the observer's meridian at the instant of obscn'ation, ^' = the reduced latitude of the place of observation, p =1 the radius of the earth for this latitude, then 0, if\ and o are the polar eo-onlinates of the observer, entirely analogous to a, o, and J of the star, so that the rectan- gular co-ordinates of the observer, taken in the first system, are a = /> cos ^''eos© b z= p ciw 9"' sin c -- p !jin y' PARALLAX. 121 and for transformation from one system to the other we have x' = x — a, !/ = y — ^f n^ = 2 — c. Hence J' cos if cos o' = J cos d cos a — p COS f ' COS J' COS ^ sin o' = J cos 5 sin a — p cos ^ sin O y (131) J' sin ^ = J sin ^ — p sin f ' or, dividing by J, and putting as before J' 1 /=_ sin»=-j- / cos ^ cos a' = COS 9 COS a — p Bin 7C COS f ' COS / COS ^ sin o' = COS ^ sin a — p sin ;r cos v>' sin y (132) / sin ^ = sin ^ — ^o sin rr sin ^' From the first two of these equations we deduce / cos ^ sin (a' — o) = /» sin TT cos f ' sin (o — 0) \ i qq / cos ^ cos (a' — a) = COS d — p Bin t: COS f' COS (a — ^) ) ^ Mnltiplj-ing the first of these by sin J (a' — a), the second by cos i (a' — a), and adding the products, we find, after dividing by cos i {a' — a), Put - V - /» sm IT cos / cos rj (tt' + o) — 01 / cos ff = cos ^ — — , ,\ ^ — ■ — ^ =i cos 1 (a' — o) . tan ^' cos 1 (a' — a) ,- „ .^ **" ^ = COB a (,- + ,) - 0] (13^) then we have, for determining 5', /sin ^ = sin ^ — p sin rr sin ^ f cos ^ = cos J — /» sin rr sin f>' cot y whence sin (^ — r) } (135) / sin (^ — ^) = P sin tt sin ^ ' sm ^ / cos (^ — ^) = 1 — p sm TT sm 0' ^ ^ sm ^ (136) J sm (^ — Y) ^ The equations (133) determine, rigorously, the parallax in right 122 PARALLAX. ascension, or a' — a ; (136) the parallax in declination, or 5' — J; and (137) determines J'. 99. To obtain the developments in series, put P sin r cos tp' m = cos d then from (133) we have J. J' f V wi sin (a — 0) ,ioo\ tan (a' — a) = = 5^-p ^. (138) ^ ^ 1 — m cos (a — 6) ^ whence m sin (a — 6) , m' sin 2 (a — O) , . ,^^^^ Putting /> sin r sin <?' n = ; n sm Y we have from (136) / V iN ^ sin (^ — y) tan (^ — d) = := ,, ^ , n40) ^ '^ 1 — n cos {^d — ;') ^.^'v-/ whence sm 1 2 sui 1 ' ^ ^ 100. The quantity a — is the hour angle of the star oast of the meridian. Acconling to the usual practice, we sliall reckon the hour angle towards the west, and denote it by /, or put t=e —a and then we shall write (138) and (140) as follows : . . ,. m sin t tan (a — o') = -— - 1 — jn cos t tan (a - J-) ="*''" "1-*^- 1 — n eos (J' — *i) Tlie rigorous computation will be conveniently performed by the followinsr forniulie: sin t> = m cos t = PARALLAX. 128 /9 sin ir cos ^' COS t COS d tan (a — a') = tan ^ tan (45® + } d) tan t tan «p' cos } (a — o') cos p + 1 (o — o')] t \ J . «/ X .N P sin iz sin «p' cos (y — S) sm ^' = n cos (j —- ^) = '- ^ ^ ^ sin Y tan (a — a') = tan ^ tan (45® + J ^) tan (y — 5) 101. Except for the moon, the first terms of the series (139) and (141) will suffice, and we may use the following approxi- mations : pn COS <p' sin t a — a = 5 COS d tan o' . ^^ ^^^ tanr = -;^ \ (143) ^ __ ^ ^ joff sin f' sin (r — ^) sin ;^ If the star is on the meridian, we have / = 0, and hence X = if'y and d — ^ = /wr sin (f ' — d) Since in the meridian we have Z = <p — 3, it is easily seen that C' — C fo^^^d by (108) and d' — d found by (140) will then be numerically equal, or the parallax in zenith distance is numeri- caUy equal to the parallax in declination loJien the star is on the meri- dian. 102. To find the parallax of a star in right ascension and declination, when its obsei^ved right ascension and declination are given. Multiplying the first equation of (132) by sin a', the second by cos a', and subtracting one product from the other, we find . , -. p sin TT cos cp' sin (0 — o') sm (a — a ) = ^^ ^ cos d In like manner, from (135) we deduce 124 PARALLAX. • /. v\ /> Bin TT sin / sin (;- — ^) sm {d — dr) = ^ sin Y We have here — a' equal to the apparent or observed hour angle ; and hence, putting t'=e — a f the computation may be made under the following form : . , ,, p sin TT cos cp' sin t' sm (a — o') = ^ ^ ^ cos a tan / cos J (tt — o') tan r = ^^ ^ cos [f — J (a — a')] . / • vv /> sin r sin ^' sin (r — ^) sm (jS — o ) = ' sin Y (144) In the first computation of a — a' we employ 5' for 8. The value of a — a' thus found is sufficiently exact for the compu- tation of -jf and o — 5'. With the computed value of 3 — d' we then find 3 and correct the computation of a — a'. Example. — Suppose that on a certain day at the Greenwich Observatory the right ascension and declination of the moon were observed to be tt' = 7* 41~ 20'.436 ^f = 15^ 50' 2r.66 when the sidereal time was e=llM7-0'.02 and the moon's equatorial horizontal parallax was n == 66' 57".5 Required the geocentric right ascension and declination. We have for Greenwich ^ = 51° 28' 3ft".2, and hence (Table III.) ^ - if' .-=- IV 13".r», ^' r. 51° 17' 24".r), log p = 9.9991134. The com- putation by (144) is then as follows: PARALLAX. 125 a' (in arc) e «< 1160 2(y 6''.54 169 15 .80 tz=z }(a-a') = </.-J(a — tt') = log 8€C [<* — J (a — log cos J (a — a') log tan ^' log tan y y = log p sin IT log sin ^' log Bin (y — d*) log cosec 7 log sin (cJ — 6') 6 — 6' = 6 = 53 54 58 .76 14 55 .8 58 89 58 o')] 0.227819 9.999996 0.096133 0.823448 = 640 85' 58" r 48 45 80 8.218877 9.892275 9.876181 0.044153 8.080986 = -f 86' 55".24 = 16O27'22".90 log p sin IT log COS ^' log sin f (1) log cos (T 8.218377 9.796142 9.907489 7.922008 9.983185 App. log sin (a — a') 7.938828 Approx. a — a' = 29* 51".6 (1) . . 7.922008 log cos 6 9.981835 log sin (a — a') 7.940173 a — o'= + 29'57".23 a =115050' 3".77 = 7A43«20'.251 103. For all bodies except the moon, the second computation will never affect the result in a sensible degree, and we may use the following approximations : , pn cos ^ sin f a — a tan^ cos ^ tan ^ cos f (m sin fff sin {y (145) -g) sin^ For the sun, planets, and comets, it is frequently more conve- nient to use the geocentric distance of the body instead of the parallax, or, at least, to deduce the parallax from the distance, the latter being given. This distance is always expressed in parts of the sun's mean distance as unity. K we put ^r^ = the sun's mean equatorial horizontal parallax, J^ = the sun's mean distance from the earth, we have, whatever unit is employed in expressing J, J,,, and a, sm TT a a sm ff„ = — J 126 PARALLAX. whence sm Tf = ~ Bin ita and when we take J^ = 1, am -K = 2. or r = -5. (146) According to Encke's detennination T^= 8".57116 log Tt^ = 0.93304 Example. — ^Donati's comet was observed by Mr. James Fer- guson at Washington, 1858 Oct. 18, 6* 26* 2r.l mean time, and its observed right ascension and declination when corrected for refraction were o! = 236^ 48' 0".5 ^ = — T*> 36' 52".8 The logarithm of the comet's distance from the earth was log A = 9.7444. Required the geocentric place. AVe have for Washington <p = 38° 53' 39". 3, whence, by Tabic in., log /o cos f ' = 9.8917, log/) sin f' = 9.7955. Converting the mean into sidereal time (Art. 50), we find = 19* 55* 16*.98. Hence, by (145) and (146), e — 298° 49'.2 log tan / 9.9038 tt' — 236 48.0 log cos f 9.6713 f = 62 1.2 log tan r 0.2325 logr„ 0.9330 r — 69*» 39'.2 log J 9.7444 r — ^= 67 16.1 logr 1.1886 log />:r cos v>' 1.0803 log pTt sin ^ 0.9841 log sin f 9.9460 log sin (r — ^) 9.9649 log sec ^ 0.0038 log eosec r 0.0640 log (a — a') 1.0801 log (a — ^) 1.0130 a — a' = + 10".7 a ~ ^ = + 10".3 Hence, for the geocentric place of the comet, a = 236° 48' 11".2 a = ~ 7° 36' 42".5 104. Parallax in ladtufk ami longitude. — Formnhe similar to the above obtain for the parallax in latitude and longitude. We REFRACTION. 127 have only to substitute for and ^' (which are the right ascension and declination of the geocentric zenith) the corresponding longitude and latitude of the geocentric zenith (which will be found by Art. 23), and put X and ^ in the place of a and d. Thus, if I and b are the longitude and latitude of the geocentric zenith, the equations (1*43) give for all objects except the moon. , pT: cos b sin (/ — X) cos ^ tan b tanr = -^ ) (147) cos (/ — A) f \ J B^ff= ^^ ^'" ^ ^'" ^^ "~ ^^ sin Y In the same manner, the equations (131) may be made to express the general relations between the geocentric and the apparent longitude and latitude, and for the moon we can employ (142), observing to substitute respectively for o, •', s, y, ©, f quantities X, >■', /9, ^, h b In all the formulfe, when we choose to neglect the compression of the earthy we have only to put ^ = ^' and p = l. REFRACTION. 105. General laws of refraction. — The path of a ray of light is a straight line so long as the ray is passing through a medium of uniform density, or through a vacuum. But when a ray passes obliquely from one medium into another of difterent density, it is bent or refracted. The ray before it enters the second medium is called the incident ray ; after it enters the second medium it is called the refracted ray; and the diiference between the directions of the incident and refracted rays is called the refraction. If a normal is drawn to the surface of the refracting medium at the point where the incident ray meets it, the angle which the incident ray makes with this normal is called the angle of inci- dence^ the angle which the refracted ray makes with the normal is the angle of refraction^ and the refraction is the difference of these two angles. 128 REFRACTION. Thus, if SAy Fig. 15, is an incident ray upon the surface BB' of a refracting medium, AC the refracted ray, 3IN the normal to the surface at A^ SAM is the angle of incidence, CAN is the angle of refraction ; and if CA be produced backwards in the direction AS\ SAS' is the refraction. An ol)ser\'er whose eve is at any point of the line AC will receive the rav as if it had come direct Iv to his eve without refraction in the direction S'ACj which is therefore called the apparmi direction of the rav. Xow, it is shown in Optics that this refraction takes place according to the following general laws: 1st. AMien a ray of light falls upon a surface (of any form) which separates two media of different densities, the plane which contains the incident rav and the normal drawn to the surface at the jK:>int of incidence contains the refracted my also. 2d. When the ray i>asses from a nirer to a denser medium, it is in general refracted toicards the normal, so that the angle of refraction is less than the angle of incidence ; and when the ray passes fn>m a denser to a rarer medium, it is refracted /row the normal, so that the angle of refraction is greater than the angle of incidence. 3il. Whatever mav be the anirle of incidence, the sine of this angle bears a constant ratio to the sine of the corri'sjKinding anirle of retraction, st> lontr as the densities of the two media are constant. If a ray passes ont i>f a vacuum into a given meflium, the number exj»ressing this constant ratio is lalled the imhx of r»fr»t**t:..n for that medium. This index is always an improper fra«-t:*»ru being c«jual to the sine of the angle oi incidence divided bv thf -iric of the auirle oi refrartion. 4::i. Wh»ii the ray passes fi\>m one medium into another, the sii;e- I'f th«- an*rlcs of incidence ami refraction art* n:*iiprocaUy ppi{«irt:oiial to the indices of refraction of the two meilia. 1<». A-^r"hom''t**if rt'fracdon, — The ravs of liirht tmm a star in Ct»miiii: t«» the obser\er must pass thn>ugh the atmosphere which sum»u!nl> the earth. If the sji;ue between the ^ta^ and the upper limit of the atmosphere be reganled as a vaeuum, or as filled with a medium which exerts no sensible effect upon the REFRACTION. 129 direction of a ray of light, the path of the ray will be at first a straight line; but upon entering the atmosphere its direction will be changed. According to the second law above stated, the new medium being the denser, the ray will be bent towards th<^ normal, which in this case is a line drawn from the centre of the earth to the surface of the atmosphere at the point of incidence. The atmosphere, however, is not of uniform density, but is most dense near the surface of the earth, and gradually decreases in density to its upper limit, where it is supposed to be of such extreme tenuity that its first eftect upon a ray of light may be considered as infinitesimal. The ray is therefore continually pass- ing from a rarer into a denser medium, and hence its direction is continually changed, so that its path becomes a curve which is concave towards the earth. The last direction of the ray, or that which it has when it reaches the eye, is that of a tangent to its curved path at this point; and the difl:erence of the direction of the ray before en- tering the atmosphere and this last direction is called the astro- nomical refraction^ or simply the refraction. Thus, Fig. 16, the ray Se from a star, entering the atmosphere at e, is bent into the curve ccA which reaches the observer at A in the direction of the tangent S'A drawn to the curve at A. If CAZ is the vertical line of the observer, or normal at A^ by the first law of the preceding article, the vertical plane of the observer which con- tains the tangent AS' must also contain the whole curve Ae and the incident ray Se. Hence refrac- tion increases the apparent altitude of a star, but does not aftect its azi- muth. The angle S'AZ is the apparent ze- nith distance of the star. The true zenith distance* is strictly the angle which a straight line drawn from the star to the point A Fig. 16. * By true xenith distance we here (and so long as we are considering only the effect of refraction) mean that which differs from the apparent lenith distance only by the refraction. Vol. L— » 130 REFRACTION. makes with the vertical line. Such a line would not coincide with the ray Se; but in consequence of the small amount of the refraction, if the line Se be produced it will meet the vertical line AZ at a point so little elevated above A that the angle which this produced line will make with the vertical will difier very little from the true zenith distance. Thus, if the produced line Se, be supposed to meet the vertical in 6', the difterence between the zenith distances measured at 6' and at A is the parallax of the star for the height Ab'y and this difference can l)c appreciable only in the case of the moon. It is therefore usual to assume Se as identical with the ray that would come to the observer directly from the star if there were no atmosphere. The only case in which the error of this assumption is appre- ciable will be considered in the Chapter on Eclipses. 107. Tables of Refraction, — For the convenience of the reader who may wish to avail himself of the refraction tables without regard to the theory by which they are computed, I shall first explain the arrangement and use of those which are given at the end of this work. Since the amount of the refraction depends upon the density of the atmosphere, and this density varies with the pressure and the temperature, which arc indicated by the barometer and the thermometer, the tables give the refraction for a mean state of the atmosphere; and when the true refraction is required, supple- mentary tables are enii)loyed which give the correction of the mean rcfniction depending upon the observed height of the barometer and thermometer. Table I. srives the rcfniction when the barometer stands at 30 inches and the thermometer (Fahrenheit's) at 50°. If we put r = the refraction, z - the ai>paront zenith distance, Z -— the true zenith distance, then : = ;: + r TVTiere great accuracy is not required, it suffices to take r directly from Table I. and to add it to z. (The resulting f is that zenith distance which we have heretofore denoted by f ' in the discussion of parallax.) The argument of this table is the apparent zenith distance z. REFRACTION. 131 Table IT. is Bessel's Refraction Table,* which is generally regarded as the most reliable of all the tables heretofore con- structed. In Column A of this table the refraction is regarded as a function of the apparent zenith distance z^ and the adopted form ofthis function is r= 0/5^7'^ tan z in which a varies slowly with the zeiyth distance, and its loga- rithm is therefore readily taken from the table with the argu- ment ^. The exponents A and X differ sensibly from unity only for great zenith distances, and also vary slowly; their values are therefore readily found from the table. The factor /9 depends upon the barometer. The actual pres- sure indicated by the barometer depends not only upon the height of the column, but also upon its temperature. It is, therefore, put under the form and log B and log 7^ arc given in the supplementary tables with the arguments "height of the barometer," and "height of the attached thermometer," respectively ; so that we have log /9 = log J? + log T Finally, log y is given directly in the supplementary table with the argument " external thermometer." This thermometer must be so exposed as to indicate truly the temperature of the atmo- sphere at the place of observation. In Column B of the table the refraction is regarded as a function of tlie true zenith distance ^ expressed under the form r = o'i9^>^'tan C and log a', A\ and /' are given in the table with the argument !^ ; /9 and x being found as before. Column A will be used when z is given to find ^ ; and Column B, when f is given to find z. Column C is intended for the computation of differential re- fraction, or the difference of refraction corresponding to small * From his AitronomUehe Untersuehungeny VoL I. 132 REFRACTION. difterences of zenith distance, and will be explained hereafter (Miorometric Observations, Vol. 11.). These tables extend only to 85° of zenith distance, bej'ond which no refraction table can be relied upon. There occur at times anomalous deviations of the refraction from the tabular value at all zenith distances; and these are most sensible at great zenith distances. Fortunately, almost all valuable astrono- mical observations can be made at zenith distances less than 85°, and indeed less than 80° ; and within this last limit we are justified by experience in placing the greatest reliance in Bessel's Table. In an extreme case, where an observation is made within 5° of the horizon, we can compute an approximate value of the refraction by the aid of the following supplement- airy table, which is based upon actual observations made by Argelander.* A|»p. zon. dibtaDcc. log Refhict. A A 85° (K 2.7G687 1.0127 1.1229 30 2.80590 1.0147 1.1408 86 2.84444 1.0172 1.1624 30 2.88555 1.0204 1.1888 87 2.08174 1.0244 1.2215 30 2.082G9 1.0298 1.2624 88 3.03686 1.0368 1.3141 30 3.09723 1.0465 1.3797 89 3.16572 1.0593 1.4653 30 3.24142 1.0780 1.5789 If we call R the refraction whose logarithm is given in this table, the refraction for a given state of the air will be found by the formula r = R?^r^ Example 1. — Given the apparent zenith distance z = 78° :50' 0", Barom. 20.770 inches. Attached Therm. — 0°.4 F., Ex- ternal Therm. — 2°.0 F. Wc find from Table H., Col. A, for 78° 30', logo 1 : 1.74981 ^l =r. 1.0032 k = 1.0328 and from the tables for barometer and thermometer, * Tabultt Regiomontantff p. 539. REFRACTION. 133 \ogB = + 0.00258 log r = + 0.04545 log T= + 0.00127 log /9 = + 0.00380 Hence the refraction is computed as follows : log a = 1.74981 A\ogfi = log iS^= + 0.00381 >l log T' = log r^ = + 0.04694 log tan z = 0.69154 r = 310^.53 = 5' 10".53 log r = 2.49210 The true zenith distance is, therefore, 78° 30' 0" + 5' 10".53 = 78° 35' 10".53. Example 2. — Given the true zenith distance ^ = 78° 35' 10".53, Barom. 29.770 inches, Attached Therm. — 0°.4 F., External Therm. — 2°.0 F. We find from Table H., Col. B, for 78° 35' 10", log a' = 1.74080 A' = 0.9967 A' = 1.0261 and from the tables for barometer and thermometer, as before, log j5 = + 0.00253 log ^ = + 0.04545 log T= + 0.00127 log /5 = + 0.00380 The refraction is then computed as follows : log a' = 1.74680 A' log /9 = log /5^' = + 0.00379 X' log r = log r^' = + 0.04663 log tan : = 0.69489 r = 310".53" = 5' 10".53 log r = 2.49211 and the apparent zenith distance is therefore 78° 30'. Example 3. — Given z = 87° 30', barometer and thermometer as in the preceding examples. By the supplementary table above given, log E = 2.98269 A = 1.0298 log /? = + 0.00380 log /3^ = + 0.00391 ;i = 1.2624 log r = + 0.04545 log r^ = + 0.05738 r= 18'26".6 logr = 3.04398 134 REFRACTION. It is important in all cases where great precision is required that the barometer and thermometer be carefully verified, to sec that they give true indications. The zero points of thermo- meters are liable to change after a certain time, and inequalities in the bore of the tube are not uncommon. A special investi- gation of every thermometer is, therefore, necessary before it is applied in any delicate research. If the capillarity of the baro- meter has not been allowed for in adjusting the scale, it must be taken into account by the observer in each reading. We may obtain the true refraction for any state of the air within 1" or 2", very expeditiously, by taking the mean refrac- tion from Table L and correcting it by Table XIV. A, and Table XIV. B. The mode of using this table is obvious from its arrangement. Thus, in Example 1 we find from Table I., Mean refr. == 4' 38".9 *< XIV. A, for Barom. 29.77, Corr. = — 2 . " XIV. B, « Therm. — 2^. " = + 32 . True rcfr. = 5' 9". which agrees with Bessel's value within l^.f). For greater accuracy, the height of the barometer should be reduced to tlie temperature 32° F., which is the standard assumed in these tables. The corrected height of the barometer in this example is 29.85, and the corresponding correction of the refraction would then be — 1"; consequently the true refraction would be 5' 10", which is only 0".5 in error. These tables furnish good approximations even at great zenith distances. Thus, we find by them, in Example 3, r = 18' 24". 108. Investigation of tue refraction for.mula. — In tliis investigation we may, without sensible error, consider the earth as a sphere, and the atmosjihere as conii»osod of an infinite number of concentric si»hcrical strata, whose common centre is the centre of the earth, each of which is of unifonn density, and within which the path <»f a ray of light is a straight line. Let (' Fig. 16, be the rentre of the earth, A a point of obsen'ation on the surface; CA Z thv vertical line ; Aa\ a'b\ tV, &c. the vertical thicknesses of the con<*entric strata; Se a my of light from a star 6', meeting the atmosphere at the point r, and suceessively re- BEFRACTION. 135 fracted in the directions erf, rfe, &c. to the point A. The last direction of the ray is aA^ which, when the number of strata is supposed to be infinite, becomes a tangent to the curve ecA at -4, and consequently AaS' is the apparent direction of the star. Let the normals C5?, CI/, &c. be drawn to the successive strata. The angle Sef is the first angle of incidence, the angle Ced the firet angle of refraction. At any intermediate point between e and -4, as e, we have Ccrf, the supplement of the angle of incidence, and 06, the angle of refraction. If now for any point, as c, in the path of the ray, we put i = the angle of incidence, / = the angle of refraction, fi = the index of refraction for the stratum above c, ax' = " " " below c, then, Art. 105, ^' = ^ (148) Bin/ fi If we put q = the normal Cc to the upper of the two strata, ^ = '' Ch '' lower « « i' = the angle of incidence in the lower stratum, = 180<^ — Che, the rectilinear triangle Cbc gives sin i' q sin/ ^ which, with the above proportion, gives q fi sin i = q'f/ sin i' an equation which shows that the product of the normal to any stratum by its index of refraction and the sine of the angle of incidence is the same for any two consecutive strata; that is, it is a constant product for all the strata. If then we put z = the apparent zenith distance, a := the normal at the observer, or radius of the earth, /i^= the index of refraction of the air at the observer, we have, since z is the angle of incidence at the observer, qfi sin i = afi^ sin z (149) 136 REFRACTION. ill which the second member is constant for the same values of z and (JL^, JS'ow, we have from (148) tan J (i -/) = ^^1^ tan } 0' +/) But i — / is tlie refraction of the ray in passing from one stratum into the next ; and supposing, as we do, that the densities of the strata vary by infinitesimal increments, i — /is the differential of the refraction ; and we may, therefore, write J dr for tan i (t — f) and dii for fi! — ijl\ consequently, also, 2/i for /i' + /jl, and tan i for tan J (i +/) : hence we have dr = ^ tan i (150) which is the differential equation of the refraction. But, as both /i and i are variable, we cannot integrate this equation unless we can express i as a function of /i. Tliis we could do by means of (149) if the relation between q and [i were given, that is, if the law of the decrease of density of the air for increasing heights above the surface of the earth were known. This, however, is unknown, and we are obliged to make an hypothesis respecting this law, and ultimately to test the validity of the hypothesis by comparing the refractions com- puted by the resnlting formula with those obtained by direct obser\'ation. I shall first consider the hypothesis of IJougi'er, both on account of the simplicity of the resulting formula and of its historical interest.* 109. First hypothesis, — Let it be assumed that the law of de- crease of density is such that some constant power of the refrac- tion index ft is recii>rocally i)roportional to the normal y, an hjliothesis expressed by the equation ♦I shall consider but two hypotheses: the first, because it leads to the simple formula of BRAnLF.Y. which, though imperfect, is often useful a:» an approximate expression of the refraction; the second, because the tables fonne<l from it bv Bess EL have thus far appeared to be the most correct and in greatest acconlance with obserration, although on theoretical grounds even the hypothesis of Hkssel is open to objection. For a review of the labors of astronomers and physicists u|H)n this difficult subject, from the earliest times to the present, sec Pie Attronomitrh^ Struhhn* brerhuny in ihrer hiitoritchen Entwiekeluug (iar</f8tell(, von Db. C. Briiins. Leipiig, 18C1. FIRST HYPOTHESIS. 137 (151) which with (149) gives sin i = / ^ jT sin zr (152) or, logarithmically, log sin I = n log fi + ^^Sy^j where the last term is constant. By differentiation, therefore. di dfi = n — tan i f^ which with (150) gives and, integrating, di dr = ^ n r = - +C n To detennine the constant Q the integral is to be taken from the upper limit of the atmosphere to the surface of the earth. At the upper limit r = ; and if we put ^ = the value of i at that limit, we have n At the lower limit the value of r is the whole atmospheric refraction, and i = z: hence r = - +0 n Eliminating the constant, we have r = '-^ (153) To find 1?, we have, by putting // = 1 in (152), since the density of the air at the upper limit is to be taken as zero, • ^ Dill S ,^ «. .^ sm »> = (154) K Having then found n^ at the surface of the earth and suitably 138 REFRACTION. determined w, we find ?? by (154), and then r by (153). The two equations may be expressed in a single formula thus : which is knoANTi as Simpson's formula, but is in fact equivalent to the fonnula first given by Bouguer in 1729 in a memoir on refraction which gained the prize of the French Academy. From (154) we find sin 2 — sin t> /i^j* — 1 sin 2 -f si*^ * /*o* + ^ whence tan i(^z — '»)= ^^—i:^ tan 1(2 + ^) and, reducing by (153), tan-r=:^^^^^ tan/r — -r\ (156) which is equivalent to Bradley's formula. If we are content to represent the refraction approximately by our fommla, we can write this in the form r = ^tan {z --ff) and we shall find, with Bradley, that for a mean state of the air corresi)onding to the barometer 20.G and thermometer 50° Fahr. we can express the observed refractions, very nearly, by taking g = 57".036, / =. 3. 110. But, as we wish our formula to represent, if possible, the actual constitution of the atmosjihere, let us endeavor to test tlio hypothesis uiK>n which it rests. In order to correspond with tlie real state of nature, it is nccc:<sf(rt/ that the constitution of the atmo- sphere trhit'h the hj/pothesis inrohrs shouhl vot onh/ agree icith the ohserrnt refraction^ but also with the height of the barometerj and with (he ob^<ernfl (fiwination of heat as the altitude of the observer above (he earth's surface increases. The discussicm of the formula will be more simple if we sul>- stitute the density of the air in the place of the index of refrac- tion. Put \ -- the density of the air at the surface of the earth, — the density of the air at any point ahove the surface. FIRST HYPOTHESIS. 139 The relation between d and /i, according to Optics, is expressed by Ai»— 1 = 4A^ (157) in which 4 & is a constant determined by experiment. Accord- ing to the experiments of Biot, 4k = 0.000588768 Since k is so small that its square will be inappreciable, we may take fi = (l+ 4kd}i = l + 2kd (158) and, consequently, /i, = l + 2H A£o- = l + 2nH and (156) becomes, still neglecting A?, tan — r = nkd^ tan 2 " i^-jr) (159) If we denote the horizontal refraction, or that for z = 90°, by r^ this formula gives tan — r« = nkd^ cot — r„ n or tan -- r,, = ^/nkd^ n and, puttmg the small arc — r^ for its tangent, i2 "We can find d^ from the observed state of the barometer and thermometer at the surface of the earth, so that in order to com- pute the horizontal refraction by this formula, for the purpose of comparing it with the observed horizontal refraction, we have only to determine the value of n. Let X = the height of any assumed point in the atmosphere above the surface of the earth, d, py g = the density and pressure of the air, and the force of grav- ity, respectively, at that point, ^0' Po9 9o = ^^® same quantities at the earth's surface. 140 BEFRACTION. At an elevation greater than x by an infinitesimal distance d!r, the pressure jp is diminished by dp. The weight of a column of air whose height is dx^ density 5, and gravity g^ is expressed by gbdx^ and this is equal to the decrement of the pressure: hence the equation </p = — gbdx By the law of gravity, we have y = ^c a« (a + xy and hence dp = — gfi^d (a + xy Now, in the hj'pothesis under consideration, we have 1 + 4A^\5^ a __l f^^'-^^^l l + ^kd V a + x^li^l ~\1 + Akdj or, neglecting the square of A, a a-^- X which gives ==l-2(n + l)A:a-0 Hence dp z=-.2g^a{ii + \)kddd Integrating, p = g^a(ii^ l)AcJ« (162) no constant being necessarj-, since p and 8 vanish together. To compare this with the observed pressure i\^ let / — - the height of a column of air of the density d^ which acted upon by the gravity g^ will be in equilibrium with the pres- sure;)^; in other words, let I be tlie hei<rht of a homogeneous atmosphere of the density 8^ which would exert the pressure p^. Tlien, by this definition, i>o = 9o\^ (163) FIRST HYPOTHESIS. 141 which with (162) gives £=(« + l)f*^ (164) At the surface of the earth, where p becomes p^ and d becomes d^ this equation gives l = (n+l)^.A:^, (165) whence i and this reduces the expression of the horizontal refraction (160) to Taking as the unit of density the value of d^ which corre- sponds to the barometer 0.76 metres and thermometer 0° C, we have, according to BiOT, 4A^o = ^-^00588768 The constant I for this state of the air is the height of a homo- geneous atmosphere which would produce the pressure 0"'.76 of the barometer when the temperature is 0° C. ; and this height is to that of the barometric column as the density of mercury is to that of the air. According to Regnault, for Barom. 0"*.76 and Therm. 0° C, mercury is 10517.3 times as heavy as air: hence we have I = 0« 76 X 10517.3 = 7993-.15 For a we shall here use the mean radius of the earth, since we have supposed the earth to be spherical, or a = 6366738 metres which gives - = 0.00125545 (167) Substituting these values in (166), we find, after dividing by sin V to reduce to seconds, r^ = 1824" = 30' 24" . But, according to Arqelander*s observations, we should have 142 REFRACTION, for Barom. O-^.TG and Therm. 0^ C, r^= 37' 31''; and the hypothesis therefore gives the horizontal refraction too small by more than 7'. 111. The hypothesis can be tested further by examining whether it represents the law of decreasing temperatures for increasing heights in the atmosphere. In the first place, we observe that m this hypothesis the densities of the strata of the atmo- sphere decrease in arithmetical j)rogression ichen the altitudes increase in arithmetical progression. For, since x is very small in compari- son with a, we have very nearly a - X a + X a and hence f = 2(n + l)H(l-^) or, by (165), X = 2i(l-^) (168) which shows that equal increments of x correspond to equal decrements of d. This last equation also gives for the upper limit of the atmo- sphere, where d = 0,x-=2l; that is, in this hypothesis the height of (he atmosphere is double that of a homogeneous atmosphere of tlic same pressure. Again, we have, by (1G4), (105), and (168), ?Ao^i==l_^ (169) p,d d, 21 ^ ^ The function ^ expresses the law of heat of the strata of the Pip atmosphere. For let r^ be the temperature at the surface of the earth, r the temperature at the height x. If the temperature were r^ in both cases, we should have ^ = - (170) Po ^0 but when the temperature is changed from r^ to r the density is diminished in the ratio 1 + « (r — - r,,) : 1, t being a constant which^ SECOND HYPOTHESIS. 143 is known from experiment; so that the true relation between the pressures and densities at different temperatures is expressed by the known formula |-=A[l+.(r-r„)] Po % whence ^ = l + e(r-ro) (171) which combined with (169) gives and hence equal increments of z correspond to equal decrements of r. Hence, m this hypothesis^ the heat of the strata of the atmo- sphere decreases as their density in arithmetical progression. The value of e, according to Rudbero and Regnault, is very nearly 1 21 — . Hence we must ascend to a height ^ = 58.6 metres, in order to experience a decrease of temperature of 1° C. But, according to the observations of Gay Lussac in his celebrated balloon ascension at Paris (in the year 1804), the decrease of temperature was 40°.25 C. for a height of 6980 metres, or 1° C. for 173 metres, so that in the hypothesis under consideration the height is altogether too small, or the decrease of temperature is too rapid. This hypothesis, therefore, is not sustained either by the observed refraction or by the observed law of the decrease of temperature. 112. Second hypothesis. — ^Before proposing a new hypothesis, let us determine the relation between the height and the density of the air at that height, when the atmosphere is assumed to be throughout of the same temperature, in which case we should have the condition (170). Resuming the differential equation (161), \a + xf pat =1 —3 a -\- X 144 REFRACTION. in which 5 is a new variable verj- nearly proportional to z. We then have dp = — g^fidds which with the supposition (170) gives ^/> _ _ 90^ Ms P Po Integrating, in which the logarithm is Napierian. The constant being determined so that p becomes p^ when 5 = 0, w^e have and therefore 1 P ^0^0 ^* log f = — ^ ^5 = — -r- Po Po where I has the value (163). Ilcnce, e being the Napierian base, I = J = e-T (172) Po \ which is the expression of the law of decreasing densities upon the supposition of a uniform temperature. In our first hypo- thesis the temperatures decrease, but nevertheless too rapidly. We nuisiy thm^ frame an hypothesis between thai and the hypothesis of a uniform temperature. Now, in our first hj-pothcsis we have by (169), >vithin terms involving the second and higher powers of 5, p^o ^ as^ and in the hypothesis of a uniform temperature, ^ = 1 Po^ The arithmetical mean between these would be P^9 _^^ ^ SECOND HTPOTHESIS. 145 buty as we have no reason for assuming exactly the arithmetical mean, Bessel proposes to take 0=e * = l_- + -(-)-&c. (178) h being a new constant to be determined so as to satisfy the observed refractions. This equation, which we shall adopt as our second hypothesis, expresses the assumed law of decreasing tempe- ratures, since, by (171), it amounts to assuming at l + f(r~r,) = c-* (174) and it follows that in this hypothesis the temperatures will not decrease in arithmetical progression with increasing heights, though they will do so very nearly for the smaller values of 5, that is, near the earth's surface. Now, combining the supposition (173) with the equation dp = — g^dds we have dp gji^ « , CL ^ ^ -£. = -^ils:ehds = eh ds P Po I Integrating and determining the constant so that for s = Oyp becomes p^, we have P» which with (173) gives* 3 = 3.e-f(«*-l)+? Inasmuch as the law of the densities thus expressed is still hypothetical, we may simplify the exponent of e. For if h is much greater than I (as is afterwards shown), we may in this ex- " as ponent put 6 * — ^ "^ T *^^ ^^ shsll have as the expression of our hypothesis d = d^e l"^A=^o« * « ^^ ' ^^ * B188BL. Fundamenta ABtronomiSy p. 2S. Vol. L— 10 146 BEFRACTION. By comparing this with (172), we see that this new hypothesis differs from that of a uniform temperature by the correction r- applied to the exponent of e. Putting, for brevity, « ^ Id /'■"•An we have d = ^ c-^ (177) in which /9 is constant. Tliis expression of the density is to be introduced into tlie differential equation of the refraction (150). Now, by (149), in which j = a + x, we liavo . . aji^ sin^r (1 — s) pt^ sin-? sin 1 =: — ^^ / '-^ (a + x)ii f ATI 1 = i iin t _ (l-a) sin z toil • v^(i — sin' I) V[?-(>- - «)* sin^ 'i = (1 — «) sin z JL r>R* * 1 1 "' \a. (1. «*^ Bin* ; ,1 From the equation /i* = 1 + 4 W we deduce and if we introduce as a constant tlic quantity 1 + \k\ (which for Barom. O-.TG and Tlicrm. 0° C. is a = 0.000294211) a — (178) —(-4) We might neglect the square of A*, and consequently, also, that of SECOND HYPOTHESIS. 147 a, with hardly appreciable error, and then this expression would become simply a — , but for greater accuracy we can retain the denominator, employing its mean value, as it varies within very narrow limits. For its greatest value, when 8 = 8^, is = 1, and its least value, when 5 = 0, is = 1 — 2a, and the mean between these values is 1 — a. Hence we shall take d/i a dd M 1 — a ^0 In the denominator of the value of tan { we have also to sub- stitute l_^=:l-_^ + ^^^ ,i\ 1 + m, Therefore, substituting in (150), we have a Bin Z (1 — 5) — dr= ^' -H^-J.) (1 — o) [coB» 2 — 2o/ 1 — ~ \ + (25 — 5») sin» z] * or, by (177), , — o/g sin 2 (l-^s) e^^'ds (1 — a) [co8»2r — 2o (1 — «-^) + (2s — s*) sin»2] i In the integration of this equation we may change the sign of the second member, since our object is only to obtain the numerical value of r. It is apparent that if we put 1 for 1 — 5 in the numerator of this expression, and also neglect the term 5* sin* 2 in the denominator, the error will be almost or quite insensible ; but, not to reject terms without examination, let us develop the expression into series. For this purpose, put the radical in the denominator under the form i/ y^ — ^ sin* Zy in which y = [cos» 2r — 2a (1 — e"^ + 25 sin* z^^ Then 1—5 _ 1 — 5 /^ __5*sin»2r\-l (y* — 5* sin* 2:) i y \ y' / 1 — 5 / ^ .5' sin* z 148 REFRACTION. 1 2sv' — s*Bin'2 . = ^ ^ &c. y 22/' Hence, restoring the value of y, we have a,p sin z e'^^ds dr = (1 — a) [cos' 2r — 2o (1 — c-^) + 28 8in» z]i ap sin z e^^sds [cos* z — 2o (1 — e"^*) + 1 5 sin* -?] (1 — a) [cos« z — 2a(l — e-^)+ 2s sin* ^]l — &c (179) We shall hereafter show that the second term of this develop- ment is insensible. Confining ourselves for the present to the first term, let us, after the method of Laplace, introduce the new variable s' such that 8 = ^ + '^(l^f^ (180) Sin* 2 tlicn this term takes the form dr = o^Bi"^^"^^^^ (181) (1 — o) [cos* z + 25' sin* 2] 1 ^ ' in which we have yet to reduce the numerator to a function of the new variable s'. Now, by Lagrange's Theorem* when * See Peirce's Curres and Fanctions, Vol. I. Art. 181. For the conyenience of the reader, howeyer, I add the following demonstration of this theorem. It is pro- posed to develop the function u :=fy in a series of aacending powers of x, z andjr being connected by the equation and the functions/and ^ being given. If from this equation y could be found at an explicit function of x and substituted in the equation ti =:/y, the deyelopment could be effected at once by Maclaurin's Theorem, according to which we should haye X 2 « = Wo + D,u^ 4- /),»Mo —-+.... + ^,"Wo-— h &c. l.z l.z....fl where Ug, D^u^ &c. denote the values of u and its successive derivatives when z = 0. It is proposed to find the values of the derivatives without recourse to the elimination of y, as this elimination is often impracticable. For brevity, put 1' = ^; then the derivatives of y = t^xT relatively to x and / are SECOND HTPOTHBSIS. 149 8 = Sf -\- af8 we have fs =/5' + -f [f^. 2)/^' ] + ^ 2) [(fO'.D/^'] + f;|;3-Z>'[(K)'.i>/y]+&c. in which/ and y> denote any functions whatever, and 2), 2)^, &c. the successive derivatives of the functions to which they are prefixed. Hence, by putting 1 — e"^' fs = e''^' fs = — —, Bin' J? this theorem gives 2>.y = r+ xDJD^y l>,y = 1 + z B^YD.y whence, eliminating z, Multiplying this by Dji^ it giyes Z>, « = TD^u (a) The deriyatiye of this equation relatively to < is This is a general theorem, whateyer function ti is of y, and consequently, also, what- eyer ftinction D^u is of y. We may then substitute in it the function Y^D^u for D^u^ and we shall haye Z>,[ Y^D^u\ = Z>, [ F» + i/),t«] (6) Now, the successiye deriyatives of (a) relatiyely to z are, by the suecessiye appli- cation of (6), making n = 1, 2, 8, &o.. But when z == 0, we haye y = t, F = ^^, and hence tfo=A />,iio = ^.2>A ... />.''tio = Z>«-i[(0O"^/O where the subscript letter of the D is omitted in the second members as unnecessary^ since t is now the only yariable. These yalues substituted in Maclaurin's Theorem giye Lagrange's Theorem : 150 BB7BA0TI0K. e-P'=e-f'' ^ ri - e-^n «-«"' fov? z *- J - , o''^ ^ [(1 - «-"•')*« -"'1 '*''—- ■D' [(1 - e -'•')* «-"•'] 1.2.3 8in«z **'^ i>-*[(l-c-^'')«(j-^''] 1. 2.3. ..nsin**;? — &c. (182) But we have in the numerator of (181) and hence, differentiating (182) and substituting the result in (181), we find dr= -I^_zdl .L-.^.'+_^2)^(l-e-^•')e-^n (1 — o) [co8« 2 + 2 8' 8in« 2r]M ^sin'^r ^^ ^ J o» +rw-.^[(i-''-''")' *'-""] + 1.2.3.. .nBin»>. -^[(^--~^*>-"'*1 + &c. I (188) To effect the differentiations expressed in the several terms of this series, we take tlic general exi)re88ion (l_e-^'')"e-^'' = (-c-^'' + l)*c-^'' where the upper sign is to be used when n is even, and the lower sign when n is odd. Differentiating this n times suceessivelyi we have i>[(l— c-^'7e-^'']=+/?»[(n+l)»c-<* + *>^''— n».n«— ^''+40.] SECOND HYPOTHESIS. 151 by means of which, making n = 1 . 2 . 3 . . . successively, we reduce (183) to the following form : (1 — a)[C08»2+2s'8in»2]il Bin«2^ ' ^1.2 8in*^^ ^ ' a»i? (184) 4- &c. > We have now to integrate the terms of this series, after having multiplied eaeh by the factor without the brackets. The inte- grals are to be taken from the surface of the earth, where 5 = 0, to the upper limit of the atmosphere ; that is, q being the nor- mal to any stratum (Art. 108), they are to be taken between the limits 5 = a and q = a + Sy H being the height of the atmo- sphere. Now, this height is not known ; but since at the upper limit the density is zero and beyond this limit the ray suffers no refraction to infinity, we can without error take the integrals between the limits q = a and j = oo , i.e. between 5 = and 5 = 1. But we may make the upper limit of s also equal to in- finity. For, by (176), ^ will not differ greatly from ^, and conse- quently will be a very large number, nearly equal to 800, as we find from (167) ; hence for 5 = 1 we have in (172) d = ^" (2.718 . .)~ which will be sensibly equal to zero, and consequently the same as we should find by taking 5 = oo . Hence the integrals may be taken between the limits 5 = and 5 = oo ; consequently, also, according to (180), between the limits 5' = and 5' = oo . Now, as every term of the series will be of the form (185) [co8» z + 2sf sin' 2:]* [cot' z + 28r\i multiplied by constants, we have only to integrate this general form. Let t be a new variable, such that cot»2f + 25' = ^ (186) 152 REFRACTION. then (185) becomes If n the integral of which is to be taken from t = \^ooiz=T (187) to < = 00 , which are the limits given by (186) for s^=0 and a' = 00 . If, therefore, we denote by '4/ {n) a function such that or 4(7i) = <j^r°'(ttc-" (188) the integral of (185) will become •^0 [cos»2+2s'8in«2:]* Vn Substituting this value in (184), making successively n = 1, 2, 8, &c., we find the following expression of the refraction: V2^ =;-^{*m + 15^ pVs) -Kl)] Bin'Z + 1.2Tsin'g E-^^+W-^*- 3 + (3) + 2*.34(2)-4(l)] + &C. I (190) which, since wc have in general 1^1.2 1.2.3 + * SECOND HTPOTHESIS. 153 can also be written as follows :"*" e •!«•* 4 (1) , ' sin*2r ^^ '^ •1/2^ / • .+ &C. 113. The only remaining difficulty is to determine the func- tion i^n), (188). In the case of the horizontal refraction, where cot ^ = and therefore also r= 0, this function becomes independent of (n), and reduces to the well-known integralf s. ~ -« v^ dte-'' = -^-^ (192) 2 * Laplace, Mieanique Ciletie, Vol. IV. p. 186 (Bowditch's Translation) ; where, however, - stands in the place of the more general symbol p here employed. This form of the refraction is due to E&, Analyse det rifraetiona tutronomiquea et ter- rettret, Strasbourg, 1799. f This useful definite integral may be readily obtained as foUows. Put k = dte — tt. Then, since the definite integral is independent of the Tariable, we doe ^^ , and, multiplying these expressions together, the order of integration being arbitrary. Let V z=iu; whence dv = idu (for in integrating, regarding v as yariable, t is regarded as constant) : then we have *»=r*r*<fM. A. <«-«a + «!«)= f^duf^dt, te-t^(^-^w») ^f^du.-—! -=}(tan-»oo — tan-»0)=^ •^0 2(l + tt») ^^ -^ 4 whence •^0 ~" 2 154 REFRACTION. where tt = 3.1415926 .... The expression for the horizontal refraction is therefore found at once by putting ^i/n for '^^ (w) in every term of (191), and sin z = lj namely: ^ 1.2.3 + &c. For small values of T, that is, for great zenith distances, we may obtain the value of the integral in (188) by a series of ascending powers of T. We have I dte-^^={ dte-^^—) dte-^^ (194) The first integral of the second member is given by (192). The second is fdte -''= C''dtll-t'+-^ ?!_+&c.\ ^0 Jo \ ^ 1.2 1.2.3 ^ / rpi 1 /TT6 1 rpt = T—— + -^'- ^— . — + &c. (195) 3 1.2 5 1.2.3 7 ^ Another development for the same case is obtained by the suc- cessive application of the method of integration by parts, as follows:* Jdte-^^ = t e-" + 2ft^dte-^^ 9 9a /• . t*dte-" ♦ By the formula yxt/v - ary ^fydz^ making always x = ? ". and dy succes- Bively = di, Odt, t\dt, &c. SECOND HYPOTHESIS. 155 whence, by introducing the limits, •^0 \ ^ 3 ^ 3.5 ^3.5.7^ / ^ ^ As the denominators increase, these series finally become con- vergent for all values of T; but they are convenient only for small values. For the greater values of Tj a development according to the descending powers may be obtained, also by the method of integration by parts, as follows :* We have Hence ^ T 2T\ 2T«(2 T«)« (2 T*)» ^ 1.3.5...(2n-l) l 1.3.5...(2n + l) r°- dt ^,» ~ (2 T*)* J"*" 2*+* ^T <*"+» The sum of a number of consecutive terms of this series is alternately greater and less than the value of the integral. But since the factors of the numerators increase, the series will at last become divergent for any value of T. Nevertheless, if we stop at any term, the sum of all the remaiimg terms will be less than this term; for if we take the sum of all the terms in the brackets, the sum of the remaining terms is _ 1.3.5...(2n + l) r" dt _„ 2» + * •^ T <"•+* * By the formulay*z dy = xy — /y dz, making always dy = t dt e " , and x BnceessiYelv = -, -.» 7:1 &c. 156 REFRACTION. The integral in this expression is evidently less than the product of the integral /, GO dt multiplied by the greatest value of e-" between the limits Tand 00 , and this greatest value is e~^. Hence the above remainder is always numerically less than 1.3.5....(2n— 1) -rr which expression is nothing more than the last term of the series (when multiplied by the factor without the brackets), taken with a contrary sign. Hence, if we do not continue the summation until the terms begin to increase, but stop at some sufficiently small term, the error of the result wall always be less than this term. Finally, the integral may be developed in the form of a con- tinued fraction, as was shown by Laplace. Putting for brevity 4(n) = Mo = — (l ^,+ -^-^-^r^ + &c.\ (198) and denoting the successive derivatives of Uq relatively to Tby u^y 2^2, &c., we have first u, = -—+ -— - -~ + &c. (199) or u, = 2Tii, — l (200) Differentiating this equation, successively, we have t/, = 2Tt/j + 2t/o W3 = 2rw, + 4i/i u^ = 2Tu^ + 6ti, &c. or, in general, t/n + i =2Tu^ + 2nUn^i n having any value in the series 1 . 2 . 3 . 4 . . . &c. SECOND HYPOTHESIS. 157 Hence we derive Un 2n or, putting * = ^ (201) 2T' fk\i »©' Mn-l 1 /*\J«, + i By (200) we have (202) «o = 2r— ^ or But £rom (202), by making n suocessirely 1, 2, 8, &c., we have «i (r .. '■'& which successively substituted in (203) give 22V. = -i^ &C.| 1 + 1 + 2A 1 + ?* 1+'* 1 + &c. (204) This can be employed for all values of ST, but when k exceeds J it will be more convenient to employ (195) or (196). The successive approximating fractions of (204) are 1 1 l + 2k 1 + bk 1 + 9A + 8A« ' z —y ^.J 1 — > &c. 1 1 + /: IJ^U 1 + 6A: + 3Af« 1 + lOit + 15A» and, in general, denoting the n** approximating fraction by — , 158 REFRACTION. fln an-^i + (n — 1) kan-2 bn "" 6„ -1 + (n — 1) kbn^2 By the preceding methods, then, the function '^n) can be computed for any value of Jl A table containing the logarithm of this function for all values of J* from to 10, is given by Bessel {Fundamenta Astronomies^ pp. 36, 37), being an extension of that first constructed by Kramp. By the aid of this table the computation of the refraction is greatly facilitated. 114. Let us now examine the second term of (179.) This term wdll have its greatest value in the horizontal refraction, when z = 90°, in which case it reduces to ape-^ Sds [I 5 — 2a (1 — «-^)] (1 — a) [25 — 2a (1 — e-^O]* Moreover, the most sensible part of the integral corresponds to small values of 5, and thereforq, since a is also very small, we may put 2a (1 — e*"^') = 2a^s. The integral thus becomes 2i(l_a)(l —afiy^O Now we have, by integrating by parts, J /9 ' 2/9 •^ and hence 2^*^o Putting fis = 7?^ this becomes, by (192), Hence the term becomes g (3 — 4a/9) \x_ 8(1— a)(l — a;9)l \2/9 SECOND HYPOTHESIS. 159 Taking Bessel's value of A = 116865.8 toises* = 227775.7 metres, and the value of I = 7993.15 metres (p. 141), we find by (176) ^ = 768.57. Substituting this and a = 0.000294211 (p. 146), the value of the above expression, reduced to seconds of arc by dividing by sin V\ is found to be only 0'^72, which in the hori- zontal refraction is insignificant This term, therefore, can be neglected (and consequently also all the subsequent terms), and the formula (191) may be regarded as the rigorous expression of the refraction. 115. In order to compute the refraction by (191), it only re- mains to determine the constants a and ^. The constant a might be found from (178) by employing the value of k deter- mined by BiOT by direct experiment upon the refractive power of atmospheric air, but in order that the formula may represent as nearly as possible the observed refractions, Bessel preferred to determine both a and ^ from observations.f Now, a depends upon the density of the air at the place of observation, and is, therefore, a function of the pressure and temperature; and ^, which involves i, also depends upon the ther- mometer, since by the definition of I it must vary with the tem- perature. The constants must, then, be determined for some assumed normal state of the air, and we must have the means of deducing their values for any other given state. Let p^ = the assumod normal pressure, T^ = " " tomporaturo, p =z the observed pressure, T = " " temperature, d^ = the normal density corresponding to p^ and t©, d = the density corresponding top and t; * Fundamenta Aitronomia, p. 40. f It should be obserred that the assumed expression of the density (177) may represent yarious hypotheses, according to the form given to j9. Thus, if we put a fi = If ^^ ^^® *^® ^<>™* (1"2) which expresses the hypothesis of a uniform tem- perature. We may therefore readily examine how far that hypothesis is in error in the horizontal refraction; for by taking the reciprocal of (167) we have in this case 3 = 796.53, and hence with a — 0.00029411 we find, by taking fifteen terms of the series (193). r^ = 39* 64''.6, which corresponds to Barom. 0^. 76, and Therm. 0*» C. This 18 2' 28".5 greater than the value given by Aboelamdicr's Observations (p. 141). Our first hypothesis gave a result too small by more than 7', and hence a true hypo- thesis must be intermediate between these, as we have already shown Arom a con* 160 BEFRACTION. then we have by (171) d = , 5> ,.L in which « is the coefficient of expansion of atmospheric air, or the expansion for 1° of the thermometer. If the thermometer is Ceniigradej we have, according to Bessel,* e = 0.0036438 From (178) it follows that a is sensibly proportional to the density, and hence if we put a^ = the value of a for the normal density 8^ we have, for any given state of the air, a, l + ^(^ — '^o) Po (206) in which for p and Pq we may use the heights of the barometric column, provided these heights are reduced to the same tem- perature of the mercury and of the scales. Again, if l^ = the height of a homogeneous atmosphere of the temperatoro Tq and any given pressure and density, then the height I for the same pressure and density, when the temperature is r, is Z = ?,[l + e(T~r„)] (206) The normal state of the air adopted by Bessel in the determi- nation of the constants, so as to represent Bradley*s obser\'a- tions, made at the Greenwich Observatory in the years 1750- 1762, was a mean state corresponding to the barometer 29.6 inches, and thermometer 50° Fahrenheit = 10° Centigrade; and for this state he found a„ = 0.000278953 sideration of the law of temperatures. At the same time, we see that the hjpothetif of a uniform temperature is nearer to the truth than the first hypothesis, and we are 80 far justified in adhering to the form 6 = d^^-fi* with the modification of substi- tuting a corrected value of 3. * This Talue, determined by Bessel, from the obserrations of stars, differs slightlj from the value ^fy more recently determined by Kudbkbq and Rkonault hj diitoi experiments upon the refractive power of the air. SECOND HYPOTHESIS. 161 or, dividing by sin 1", ao = 57" .538 and h = 116865.8 toises = 227775.7 motros. For the constant l^ at the normal temperature 50° F., Bessel employed l^ = 4226.05 toises = 8236.73 metres.* Since the strata of the atmosphere are supposed to be parallel to the earth's surface, Bessel employed for a the radius of curva- ture of the meridian for the latitude of Greenwich (the observa- tions of Bradley being taken in the meridian), and, in accordance with the compression of the earth assumed at the time when this investigation was made, he took a = 6372970 metres. Uence we have A = ^-^" • - = 745.747 These values of a^, and ^„ being substituted for a and ^ in (193), the horizontal refraction is found to be only about V too great, which is hardly greater than the probable error of the obser\^ed horizontal refraction. At zenith distances less than 85°, however, Bessel afterw-ards found that the refraction com- puted with these values of the constants required to be multi- plied by the factor 1.003282 in order to represent the Konigsberg obser\'ations. 116. By the preceding formulfe, then, the values of the con- stants a and /9 can be found for any state of the air, as given by the barometer and thermometer at the place of observation, and then the true refraction might be directly computed by (191). But, as this computation would be too troublesome in practice, the mean refraction is computed for the as^med normal values of a and ^, and given in the refraction tables. From this mean * According to the later determination of Regnault which we hare used on p. 143, we should have /o=S2S6.1 metres. The difference does not affect the value of Bbssbl's tables, which are con3tructed to represent actual obserrations. Vol. L— 11 162 BEFRACTIOX. refraction we must deduce the true refraction in any case by applying proper corrections depending upon the observed state of the barometer and thermometer. For &cility of logarithmic computation, Bessel adopted the form r = r.ltYi "^ Y (207) in which r, is the tabular refraction corresponding to p^ and r^ and r is the refraction corresponding to the observed p and r. Let us see what interpretation must be given to the exponents A and X. If the pressure remained p^ the refraction correspond- ing to the temperature r would be , dr . ^ , rfV (r — T.)« , . or, with sofficient precision, '•o {■+lf<'-'^} In like manner, if the temperature were constant, and the pres- sure is increased by the quantity p — p^ the refraction would become nearly Ilcncc, when both pressure and temperature vary, we shall have, ver}' nearly, Xow, putting — in (207) under the form 1 + — --^,anddevelop- Po Po ing by the binomial theorem, we have r = r, 1 1 + - (;> -;>o) + &c. | X { 1 -^e (r-rj + 4e. | Therefore, neglecting the smaller terms, we must have A = ^.'Ji ^ = --T (209) SECOND HYPOTHESIS. 163 to determine which we are now to find the derivatives of (191) relatively to jp and r. Put X = -^ (210) Bin* z ^ ^ and ?! = 1^ (1), Ja = 2^(2), g, = 3^(3), &c., or in general ff. = n'*« 4(n) (211) then, if we also put q = xe-^q,^^e-^-q, + ^ ^ "^ ^ ^-""g^H- &c. (212) the formula (191) becomes (l-a)r=Bin«2^.e (213) in which, since the variations of ^— — in (191) are sensibly the same as those of a, we may regard 1 — a as constant. Differen- tiating this, observing that Q varies with both j? and r, while ^ varies only with r, we have (1 — o) — = sm* z \ -^ ^ ^ dp \fi dp ) (214) dr \^ dr "^ "^2^ "d^ ^ dr \ R d-r ^ ^9 A In differentiating §, it will be convenient to regard it as a func- tion of the two variables x and ^, the quantities y^ jj, &c. vary- ing only with ^. We have, since ^ does not vary with p, dQ^dQ dx ^215) dp dx dp and since both x and Q vary with r, From (212) we find 164 KEFRACTION, in which «' = ^^"' ffi + j^ e"'' 2q, + ^-^ 6-3. 3j^ + &c. (218) Also, ^ = a:e-ii' + -^e-^^' + &c. (219) d/9 <f/9 ^1.2 da ^ ^ ^ in which we have generally, by (211), But by (200), in which ?«, = i//(n), we have and by (187) dT -^*- whence cot* 2 cot Z = ng — 2 i/2^" n N Snbstitntinn^ the values of this expression for ?i = 1, 2, 3, &c. in (219), we have The first series in this expression — Q^. The second, when e~'j t' " '^, &c. are developed in series, becomes X x -!- x' 4- a:* -f &c. = J "^ I l—x SECOND HYPOTHESIS. 165 and hence § = ?2^1q'^J2^.^ (220) We have, further, from (210) and the values of a, I, and ^ in the preceding article, dx X da X a X dp a dp~~ a p """^ dfi d? dl a dr ~~ dl dr" Z« ' ^ ^ ~" h — l da dr dx X da X dp 2h — I dr a dr ' p dr h — l Substituting these values in (215) and (216), and then substituting in (214), we find* .^ .dr . - /2 ^, \—x These formulse are to be computed with the normal values of a, ^, r, I, and p^ and for the different zenith distances, after which A and X are computed by (209). The values of A and 3i thus found are given in Table 11. 117. Finally, in tabulating the formula (207), Bessel puts r^ = a tan z (222) P = A r = l + '-(^~^o) (where a and )9 no longer have the same signification as in the preceding articles). ^—~- — ■ * — * Bkssel, Fundamenta Attronomia^ p. 34. 166 BEFRACTION. The true refraction then takes the form r = aP^r^ tan z (223) The quantity here denoted by /J is the ratio of the observed and normal heights of the barometer, both being reduced to the same temperature of the mercury and of their scales. First, to correct for the temperature of the scale, let b^^y b^^\ or 6^"*> denote the ob- served reading of the barometer scale according as it is graduated in Paris lines, English inches, or French metres. The standard temperatures of the Paris line is 13° Reaumur, of the English inch 62° Fahrenheit, and of the French metre 0° Centigrade ; that is, the graduations of the several scales indicate tnie heights only when the attached thermometers indicate these temperatures respectively. The expansion of brass from the freezing point to the boiling point is .0018782 of its length at the freezing point If then the reading of the attached thermometer is denoted either by r',/', or c', according as it is Keaumur's, Fahrenheit's, or the Centigrade, the true height observ^ed will be (putting s = 0.0018782) l + -r' 1+— (/' — 32) i + JL.c' 6(0. '' , 6(->> ''' \ 60^). ;^ 1 + -.13 l^_-^.30 ^ 'so 180 or 80+^ J,, 180_+_(r^J2)£ ^(.) 100 + c-. 80 + 135' 180 + 306* 100 ^ ^ where the multipliers 1 + — r', &c. evidently reduce the reading to what it would have been if the observed temperature had been that of freezing, and the divisors 1 + — • 13, &c. further reduce those to the respective temperatures of graduation, and conse- quently give the true heights. This true height of the mercury will be proportional to tlie j)ressure only when the temperature of the mercury is constant. AVo must, therefore, reduce the height to what it would be if the temperature were equal to the adopted normal temperature, wliich is iu our table 8° Keaumur ^-^ 50° F. =^ 10° C. Now, mercury expands --- of its volume at the freezing point of water, when *' 65.6 SECOND HYPOTHESIS. 167 its temperature is raised from that point to the boiling point of water. Hence, putting q = — , the above heights will be reduced to the normal temperature by multiplying them respectively by the factors SO+Sq 180 + 18g 100 + lOg ,^26^ 80 + rV l80+(/' — 32)j' 100 + c'j ^" ^ The normal height of the barometer adopted by Bessel was 29.6 inches of Bradley's instrument, or 333.28 Paris lines ; but it after- wards appeared that this instrument gave the heights too small by J a Paris line, so that the normal height in the tables is 333.78 Paris lines, at the adopted normal temperature of 8° R. Reducing this to the standard temperature of the Paris line = 13° R., we have In comparing this with the observed heights, the 6^*^ and 6^"*^ must be reduced to lines by observing that one English inch = 11.2595 Paris lines, and one metre = 443.296 Paris lines. Making this reduction, the value of j(9 = - is found by dividing the product of (224) and (225) by (226). The result may then be separated into two factors, one of which depends upon the observed height of the barometric column, and the other upon the attached ther- mometer ; so that if we put jB = ^»('> SO + Sq 333.78 80 + 85 — h(e) 11-25^^ 80 + 13^ 180 + 18g "~ ' 333.78 ' 80 -f~87 ' 180 + 30s — 5(«) ^^-^^^ 80 + Us 100 + lOg ^ (^^^^ ■" * 333.78 ' 80 4- 85 ' 100 and T _ 80 + r's _ 180 + (f — 32)s _ 100 + c's "~ 80 + r*q ~" 180 -}-(/' — 32) ^ ~ 100 + c'q we shall have ^ = BTj or log /9 = log iJ + log T (228) 168 REFRACTION. The quantity y would be computed directly under the form r = 1 + Kt-t,) if To were at once the freezing point and the normal temperature of the tables ; for e is properly the expansion of the air for each degree of the thennometer above the freezing point, the density of the air at this point being taken as the unit of density. But if the normal temperature is denoted by r^,, that of the freezing point by r^, the observed by r, we shall have l+c(r, -rO l_l.e(r-T,) an expression which, if we neglect the square of e, will be reduced to the above more simple one by dividing the numerator and denominator by 1 + £(7^ — Tj). Bessel adopted for r^ the value 50° F. by Bradley's thermometer; but as this thermometer was found to give 1°.25 too much, the normal vahie of the tables is r^ = 48°.75 F. Hence, if r,/, or c denote the temperature indi- cated by the external thermometer, according as it is Reaumur, Fahr., or Cent, we have* 180 + 16.75 X 0.36438 ^ ■" 180 + J r X 0.36438 — 180 + 16 .75 X 0.36438 "" 180 + (/ — 32)" X 0.36438 / (^29) _ J180 + 16.75 X 0.36438 "" ~i80"4- |<^ X 0^6438 The tables constructed according to these formulie give the values of log 5, log T^ and log y^ with the arguments barometer, attached thermometer, and external thermometer respectively, and the eonii)utation of the true refraction is rendered extremely simple. An example has already been given in Art. 107. 118. In the preceding discussion we have omitted any con- sideration of the hygrometric state of the atnios]>here. The * TabuUe Regiomontantty p. LXII. REFRACTION. 169 refractive power of aqueous vapor is greater than that of at- mospheric air of the same density, but under the same pressure its density is less than that of air ; and Laplace has shown that " the increase of the greater refractive power of vapor is in a great degree compensated by its decrease of density/** 119. Refraction (able iciih the argument true zenith distance. — ^When the true zenith distance ^ is given, we may still find the refrac- tion from the usual tables, or Col. A of Table 11., where the apparent zenith distance z is the argument, by successive ap- proximations. For, entering the table with ^ instead of Zy we shall obtain an approximate value of ?•, which, subtracted from i^, will give an approximate value of z ; with this a more exact value of r can be found, and a second value of z, and so on, until the computed values of r and z exactly satisfy the equation z = ^ — r. But it is more convenient to obtain the refraction directly with the argument ^. For this puq^ose Col. B of Table 11. gives the quantities a', j1', ^', which are entirely analogous to the a. Ay and >l, so that the refraction is computed under the form r = a'y9^>A' tan C (230) where /9 and y have the same values as before. The values of a', ^', and ^' are deduced from those of a. Ay and >l after the latter have been tabulated. They are to be so determined as to satisfy the equations a,3^r^ tan z = a'^^'r^' tan C (231) J = C — a'^S^V^' tan C (232) and this for any values of ^ and x- Let {z) denote the value of z which corresponds to ^ when /9 = 1, y = 1\ that is, when the refraction is at its mean tabular value. The value of {z) may be found by successive approximations from Col. A,, as above ex- plained. Let (a), {A)y (/), and (r) denote the correspondhig values of a. Ay ly r. We have (r) = (a) tan (z) = a tan C Iz) = Z — a tan C whence, by (232), ♦ Mie, CH. Book X. 170 REFRACTIOX. r = (^) — a! tan C (;5^>^' — 1) But, taking Napierian logarithms, we have and hence, e being the Napierian base, /S^>A' = e ^'^^ + A''r = 1 + (A' l? + k' Ir) + &c. where, as /9 and ;* differ but little from unity, tlie higher powers of A'l^ + k'ly may be omitted. Hence z = {z) — (r)iA'ip + X^lr'] Now, taking the logarithm of (231), we have I (a tan z) + Alp + Xlr = l (a' tan C) + A'lp + I'ly The first member is a function of z^ which we may develop as a function of (z) ; for, denoting this first member by/?, and putting y = - (r) [A'//3 + ^7r] we have z = (z)+ y> ^^^ hence fz =f iiz) + y] =/ {z) + ^^^ y + &c., where we may also neglect the higher powers of y. But since f{z) is what/r becomes when z = (z)y and consequently -4 = (-4), / — {X\ we have fir) = I [(O tan (^)] + (4) I? + (^) /r d^jTfr)^ _ dl[iia) tan a-)] _ d [(a\ tan^)]_ ^ _1_ J<r) d(z) ~~ ci (c) "" (a) tan (c) t/ (-:) " (r) ii(c) Hence we have fz = . I [fa) tan (r)] + (.1) //? + (X) /r - Jj[^^] [4' //5 + x' /r] ^:/[o'tan:] + A'l? -\-yir or, since (a) tan [z) ~ a' tan J, REFRACTION. 171 Since this is to be satisfied for indeterminate values of ^ and y^ the coefficients of l^ and ly in the two members must be equal; and therefore / = rf(r) ) (233) and also a' = (a)*'^"(^> tan C All the quantities in the second members of these formulae may be found from Column A of Table 11., and thus Column B may be formed.* If we put we shall now find the refraction under the form r = k' tan C 120. To find the refraction of a star in right ascension and decli- nation. The declination d and hour angle t of the star being given, together with the latitude (p of the place of observation, we first compute the true zenith distance ^ and the parallactic angle q by (20). The refraction will be expressed under the form r = k' tan C in which The latitude and azimuth being here constant (since refrac- tion acts only in the vertical circle), we have from (50), by put- ♦ See also Bessel, Astronomisehe Unterauchunyen^ Vol. I. p. 159. 172 BEFRACTIOK. ting rf^ = 0, rf^ = 0, df = r = A' tan r^di = — dd, (a = star's right ascension), d9 = cos dda = — k' tan C cos q — /:' tan C sin q } (234) which are readily computed, since the logarithms of tan ^ cos q and tan ^ sin q will already have been found in computing ^ by (20). The value of log /j' will be found from Table 11. Column B, with the argument ^. The values of do and da thus found are those which are to be algebraically added to the apparent declination and right ascen- sion to free them from the eftect of refraction. The mean value of A' is about 57", which may be employed when a very precise result is not required. Fig. 17. DIP OF THE HORIZON. 121. The dip of the horizon is the angle of depression of the visible sea horizon below the true horizon, arising from the ele- vation of the eye of the observer above the level of the sea. Let CZy Fig. 17, be the vertical line of an obser\'er at -4, whose height above the level of the sea is AB, The plane of the true ho- rizon of the observer at A is a plane at right angles to the vertical line (Art. 3). Let a vertical plane bo passed through CZ, and let BTD bo the intersection of this plane with the earth's surface regarded as a sphere. All its intersection with the horizon- tal plane. Draw ATIP in this plane, tan«cent to the circular section of the earth at T, Disregarding for the pre- sent the ettect of the atmosi>here, T'will be the most distant point of the surfai'C visible fn>m A, If we now conceive the vertical plane to revolve about ("^^as an axis, AH will generate the ]»lane of the celestial horizon, while AW will generate the surface of a cone touching the earth in the small circle called the visible horizon; and the angle HAH' will be the dip of the horizon. DIP OF THE HORIZON. 173 122. To find the dip of the horizortj neglecting the atmospheric refrac- tion. Let X = the height of the eye = AB, a = the radius of the earth, D = the dip of the horizon. We have in the triangle CAT, ACT= HAH' = D, and hence tanD = -— -- CT By geometry, we have AT=VAB X AD = Vx{2a-{'X) whence a ^ a ' \ a / As a: is always very small compared with a, the square of the fraction — is altogether inappreciable: so that we may take simply tan i) = J^ (235) 123. To find the dip of the horizon, having regard to the atmospheric refraction. The curved path of a ray of light from the point T, Fig. 18, to the eye at A, is the same as that of a ray from A to T; and this is ^l^' ^^* a portion of the whole path of a ray (as from a star S) which passes through the point A, and is tangent to the earth's surface at T The ^' direction in which the observer at h' A sees the point T is that of the tangent to the curved path at A, or AH'; the true dip is therefore the angle HAH', and is less than that found in the preceding article. It is also evident that the most distant visible point of the earth's 174 DIP OP THE horizon; surface is more remote from the observer than it would be if the earth had no atmosphere. Now, recurring to the investigation of the refraction in Art 108, we observe that the angle HAH' is the complement of the angle of incidence of the ray at the point A^ there denoted by i; and it was there shown that if y, [Xy and i are respectively the normal, the index of refraction, and the angle of incidence for a point elevated above the earth's surface, while a, /i^ and z are the same quantities at the surface, we have • • 9 /£ sm t = a A<o sm ^ But in the present case we have z = 90° ; and hence, putting D' = the true dip = 90^ — i q =a -{- X we have sin t = cos iJ = — . = — 1 1 H I M a -\- X m\ a I X Developing and neglecting the square of — as before, cosi)' = ^(l-|^) (236) which would suffice to determine D' when /^ and fi have been obtained from the observed densities of the air at the observer and at the level of the sea. But, as JD' is small, it is more con- venient to determine it from its sine; and we may also intro- duce the density of the air directly into the formula by putting (Art. 110), M \ 1 + 4^^ Substituting the value of a from (178), namely, _ 2kdo * ""1 + 4^^0 we may give this the form DIP OF THE HORIZON. 175 ={i-Mi-:^} 8 1-i which, by neglecting the square of the second tenn, gives ^=> + -('-t) Hence, still neglectiug the higher powers of a and — , as well as their product, we have 8mi>' = >/l— co8*i>' = ^|^— 2a/l— ^U (237) which agrees with the formula given by Laplace, JUffc. CSl. Hook X. For an altitude of a few feet, the difference of pressure will not sensibly affect the value of i)', and may be disregarded, especially since a very precise determination of the dip is not possible unless we know the density of the air at the visible hori" Z071, which cannot usually be observed. We may, however, assume the temperature of the water to be that of the lowest stratum of the air, and, denoting this by r^, while r denotes the temperature of the air at the height of the eye, we have [mak- ings = Pq in (171)], approximately, in which for Fahrenheit's thermometer e = 0.002024. Hence ^sinDJl-^I^Ill^H I sin' D ) where D is the dip, computed by (235), when the refraction is neglected, the sine of so small an angle being put for its tan- gent. If we substitute the values a = 0.00027895, sin D = D sin 1", and $ = 0.002024, this formula becomes 176 DIP OF THE HORIZON. _pr_j 24021 (r~r,) in which D is in seconds. If JD is expressed in minutes in the last term, it will be sufficiently accurate to take 2)' = 2)-400x^^ (238) This will give D' = D when r = r,,, as it should do, since in that case the atmosphere is supposed to be of uniform density from the level of the sea to the height of the observer. If ^ < ^o» ^'® have D' '^ D. In extreme cases, where r is much greater than r^,, we may have JD' < 0, or negative, and the visible horizon will appear above the level of the eye, a phcnomenou occasionally observed. I know of no observations sufficiently precise to determine whether this simple formula, deduced from theoretical considerations, accurately represents the observed dip in every case. 124. If, however, we wish to compute the value of D' for a mean state of the atmosphere without reference to the actually observed temperatures, we may proceed as follows : In the equa- tion above found, cos2)'= - . we may substitute the value //I \" + ^_ a \/iJ ~a+x which is our first hypothesis as to the law of decrease of density of the strata of the atmosphere, Art. 109. This hypothesis will serv'c our present i)urpo8e, provided 7i is so determined as to rei)re8cnt the actually observed mean horizontal refraction. We have, then. =('+i)- and developing, neglecting the higher powers of—, DIP OF THE HORIZON. 177 n X COS D' = 1 — n + 1 a Bin D' = \l — r-T • — = tan D \l — ^T \n4-l a \n + 1 + 1 a \n + or *'=^Vir^ + To determine n, we have by (160), reducing Vq to seconds, n = (ro sin 1")* where, for Barom. 0*.76, Therm. 10°C., which nearly represent the mean state of the atmosphere at the surface of the earth, we have 4k d^ = 0.00056795, and r^ = 84' 30" (which is about the mean of the determinations of the horizontal refraction by dif- ferent astronomers) ; and hence we find n = 5.639, J-JL. = 0.9216 = 1 — 0.0784 \n + 1 D' = D — .0784i) (239) The coefficient .0784 agrees very nearly with Delambre's value .07876, which was derived from a large number of observations upon the terrestrial refraction at different seasons of the year. To compute D' directly, we have sm 1" ^ d If a: is in feet, we must take a in feet. Taking the mean value a = 20888625 feet, and reducing the constant coefficient of >/F, we have D' = 58".82 i/x in feet. (240) Table XI., Vol. IL, is computed by this formula. Vol. L— 12 178 DISTANCE OF THE HORIZON. 125. To find the distance of the sea horizon^ and the distance of an object of known height jxist visible in the horizon. — The small portion TAy Fig. 19, of the curved path of a ray of ^**' ^^' ligl^t, may be regarded as the arc of a circle; and then the refraction elevates A as seen from T as much as it elevates T as seen from A. Drawing the tangent TP, the ob- server at T would see the point A at P; A'^Bf \ \ ai^j if the chord TA were drawn, the angle PTA would be the refraction of A. This refraction, being the same as that of T as seen from A^ is, by (239), equal to .07842). In the triangle TPAy TAP is so nearly a right angle (mth the small elevations of the eye here considered) that if we put x^ = AP we may take as a sufficient approximation x^ = TAy^ tan PTA = a tan D X .0784 tanD But we have a tan^JD = 2a:, and hence x^ = .1568 a; Putting d = the distance of the sea horizon, we have PT=y'i2CB + PB) X PB or, nearly, d = \/2a (.r + .rj = v/2.3136aa: If X is given in feet, we shall find d in statute miles by dividing this value by 5280. Taking a as in the preceding article, we find 5280 and, therefore. V2Mma _ j_3j^ d (in statute miles) t= 1.317 |/x in I'ect. (241) If an ol)server at -4' at the height A'B' — x' sees the object -4, whose height is a:, in the horizon, he must be in the curve do- DIP OF THE SEA. 179 scribed by the ray from A which touches the earth's surface at T. The distance of A* from Twill be = 1.317 i/x^', and hence the whole distance from A to A^ will be = 1.317 {Vx + Vx^). Tlie above is a rather rough approximation, but yet quite as accurate as the nature of tlie problem requires ; for the anoma- lous variations of the horizontal refraction produce greater errors than those resulting from the formula. By means of this formula the navigator approaching the land may take advantage of the first appearance of a mountain of known height, to deter- mine the position of the ship. For this purpose the formula (241) is tabulated with the argument "height of the object or eye ;" and the sum of the two distances given in the table, cor- responding to the height of the object and of the eye respect- ively, is the required distance of the object from the observer. 126. To find the dip of the sea at a given distance from the observer. — ^By the dip of the sea is here understood the apparent depres- sion of any point of the surface of the water nearer than the visible horizon. Let T, Fig. 20, be such a point, and A the position of the observer. Let TA' be a ray of light from T, tangent to the earth's surface at T, meeting the ver- tical line of the observer in A'. Put D"= the dip of T sls seen from A, d = the distance of T in statute miles, X = the height of the observer's eye in feet = AB^ x' = A'B. We have, by (241), -=(r^r and the dip of Ty as seen from A'^ is, therefore, by (240), = 58".82 i/F = 44".66 d. Now, supposing the chords 7!A, TA' to be drawn, the dip of T at A exceeds that at A' by the angle A TA'^ very nearly ; and we liave nearly aiii'lo ATA = X = TA' sinl" 5280 <« sin 1" 180 SEMIDIAMETERS. whence 5280 d Bin 1" Substituting the value of x' in terms of rf, D" = 22'M4 d + 39".07 - (x being in feet and d in statute miles). (242) 691 If rf is given in sea miles, we find, by exchanging d for 7~rf, D" = 25".65 d + 33".73 -(x being in feet and d in sea d miles). (243) ■ The value of D" is given in nautical works in a small table with the arguments x and d. The formula (243) is xQty nearly the same as that adopted by Bowditch in the Pi^actical Navigaicr. 127. At sea the altitude of a star is obtained by measuring its angular distance above the visible horizon, which generally appears as a well-defined line. The observed altitude then exceeds the apparent altitude by the dip, remembering that by apparent altitude we mean the altitude referred to the true horizon, or the conipleinent of the apparent zenith distance. Thus, A' being the observed altitude, h the apparent altitude, or, when the star has been refeiTcd to a point nearer than the visible horizon, PEMIDIAMETERS OF CELESTIAL BODIES. 128. In order to obtain by observation the position of the centre of a celestial body which has a well-defined disc, we observe the position of some point of the limb and deduce that of the centre bv a suitable application of the angular scmi- diametcr of the bodv. I shall here consider onlv the case of a si)herical bodv. The ft I • api^arent outline of a jilanct, whether s]»herical or spheroidal, and whether fully or partially illuminated by the sun, will be SEMIDIAMETERS. 181 Fig. 21. discussed in connection \vith the theor}' of occultations in Chapter X. The angular semidiametcr of a spherical body is the angle subtended at the place of obsei-v^ation by the radius of the disc. I shall here call it simply the seniidiameter, and distinguish the linear seniidiameter as the radius. Let O, Fig. 21, be the centre of the earth, A the position of an ob- server on its surface, M the centre of the observed body; 0J5, AB'^ tangents to its surface, drawn from and ^. The triangle OEM re- volved about OM as an axis will de- scribe a cone touching the spherical body in the small circle described by the point J5, and this circle is the disc whose angular seniidiameter at is MOB. Put S = the geocentric semidiametcr, MOB, S' = the apparent semidiametor, MAB\ J,J' = the distances of the centre of the body from the centre of the earth and the place of observation respectively, a = the equatorial radius of the earth, a' = the radius of the body, then the right triangles OMBy AMB' give . „ a! sm o = — J sin S' = a' (244) But if T = the equatorial horizontal parallax of the body, we have, Art. 89, and hence a SHI TT = — J sm aS = — sm t: a sin S' = — sin S J' (245) or, with sufficient precision in most cases, a S'= — 8 J' (246) 182 SEMIDIAMETERS. The geocentric semidiameter and the horizontal parallax have a' therefore a constant ratio = — . For the moon, we have a a' - = 0.272956 (247) a a8 derived from the Greenwicli observations and adopted by Hansen {Tables de la Lwie, p. 39). If the body is in the horizon of the observer, its distance from liim is nearly tlic same as from the centre of the earth, and hence tlie geocentric is frequently called the horizontal semidiameter; but this designation is not exact, as the latter is somewhat greater than the former. In the case of the moon the difterence is betAveen 0".l and 0".2. See Table XII. If the body is in the zenith, its distance from the observer la less than its geocentric distance by a radius of the earth, and the apparent semidiameter has then its greatest value. The apparent semidiameter at a given place on the earth's surface is computed by the second equation of (245) or (246), in which the value of — is that found by (104) ; so that, putting 2 — the true (geocentric) zenith distance of the body, ^' = the appa- rent zenith distance (affected by parallax), A — its azimuth, f — <p' the reduction of the latitude, we have, (by (111) and (104), Y =. {<p — if') Qo?^ A \ 8in(C —r) ^ 129. This last formula is rigorous, but an approximate fonnnla for computing the difference aV — aS will sometimes be convenient. In (103) we may put cos (i p — if') _ cos /'COS ] (C' — C) without sensible error in computing the very small diflerence in question ; we thus obtain J' ^, 1 _ ^ sin - cos [} (C 4- :) — f\ SEMIDIAMETERS. 183 Putting m = p&mi: COS [i (C' + C) — r] (249) we have J 1 J' 1 — m = 1 + m + m* + &o. and hence, smce the third power of m is evidently insensible, S' — S = Sm + &n» (250) which is practically as exact as (248). The value of ^' required in (249) will be found with sufficient accuracy by (114), or C' — C = /o7r8in(C' — r) The quantity S' — S is usually called the augmentation of the semidiameter. It is appreciable only in the case of the moon. 130. If we neglect the compression of the earth, which will not involve an error of more than 0".05 even for the moon,* we may develop (250) as follows. Putting /> = 1 and 7* = in (249), we may take m = sin ff cos i (C + C) = sin n cos [C — } (c' — 0] = sin TT cos C' + } sin tt sin (C' — C) sin C = sin n cos C' + i sin'r sin' C' which substituted in (250) gives, by neglecting powers of sin n above the second, /S' — S = iS sin TT COS C + i S sin^w sin' C + S sin'Tr cos* C = 8 sin TT cos C' + }t S sin'Tr -\- k S sin*7r cos* C' But we have „ a' a* sin it « a sin 1" * The greatest declination of the moon being less than 30°, it can reach great altitudes only in low latitudes, where the compression is less sensible. A rigorous ioTestigation of the error produced by neglecting the compression shows that the maximum error is less than 0".06. 184 SEMIDIAMETEBS. and if we put A = iL sin 1", log h = 5.2495 we have sin t: ^ hS, which substituted above gives the follow- ing formula for computing the augmentation of the moon's semidiameter: S'—S = hS'cos:'+ih^S*+ JA«/S»cos«C' (251) Example.— Find the augmentation for C' = 40°, S = 16' 0" = 960". Ist term = 12^.54 2d « = .14 3d " = 0. 08 iS' — S = 12 .76 log 3d term 8.914 The value of S' — S may be taken directly from Table XTE. with the argument apparent altitude = 90® — ^'. 131. If the geocentric hour angle (t) and declination {d) are given, we have, by substituting (137) in (245), sin S' = sin S ^1" ^// ~ ^} (252) sm (^ — y) ^ ' for which y and 5' are to be determined by (134) and (136), or with sufficient accuracy for the present purpose by the formulae tan <p' ' cos t log /S« 5.9645 log S^ 8.947 log h 5.2495 log J A« 0.198 log cos C 9.8843 log 2d term 9.145 log Ist term 1.0983 log cos' C 9.769 ^ — ^ = p n sin ^' sin (d — y) sin ^ 132. To find the contraction of the vertical semidiameter of the sun or moon produced by atmospheric refraction. Since the refraction incroases with the zenith distance, the refraction for the centre of the sun or the moon will be greater than that for the upper limb, and that for the lower limb will be greater than that for the centre. The apparent distance of the SEMIDIAMETERS. 185 limbs is therefore diminished, and the whole disc, instead of being circular, presents an oval figure, the vertical diameter of which is the least, and the horizontal diameter the greatest. The refraction increasing more and more rapidly as the zenith distance increases, the lower half of the disc is somewhat moro contracted than the upper half. The contraction of the vertical semidiameter may be found directly from the refraction table, by taking the difterence of the refractions for the centre and the limb. Example. — The true semidiameter of the moon being 16' 0", and the apparent zenith distance of the centre 84°, find the con- traction of the upper and lower semidiameters in a mean state of the atmosphere (Barom. 30 inches, Therm. 50° F.). We find from Table I. For apparent zcn. dist. of centre, 84° 0' Eefr. = 8' 28".0 " approx. « upper limb, 83 44 « = 8 9 .4 " " " lower " 84 16 " =8 48 .1 Ilence, Approx. contraction upper semid. = 8' 28".0 — 8' 9".4 = 18".6 " " lower " = 8 48 .1 — 8 28 .0 = 20 .1 These results are but approximate, since we have supposed the apparent zenith distance of the limb to difter from that of the centre by the true semidiameter, whereas they difter only by the apparent or contracted semidiameter. Ilence we must repeat as follows : App. zen. dist. upper limb = 83° 44' 18".6 Eefr. = 8' 9".7 " " lower " = 84 16 39 .9 " = 8 47 .7 Contraction of upper semid. = 8' 28".0 — 8' 9".7 = 18".3 " lower " = 8 47 .7 — 8 28 .0 = 19 .7 Obser\'ations at great zenith distances, where this contraction is most sensible, do not usually admit of great precision, on account of the imperfect definition of the limbs and the uncer- tainty of the refraction itself. It is, therefore, sufliciently exact to assume the contraction of either the upper or lower semi- diameter to be equal to the mean of the two. In the above example, which oftei's an extreme case, if we take the mean 186 SEMIDIAMETERS. 19" as the contraction for either seniidiameter, the error vnll be only 0".7, which is quite within the limit of error of observa- tions at such zenith distances. 133. To find the contraction of any inclined seniidiameler^ produced by refraction. Let My Fig. 22, be the apparent place of the sun's or tlie moon*s centre; ACBD^ a circle described with a radius 3IA equal to the true semi- diameter, will represent the disc as it would appear if the refraction were the same at all points of the limb. The jKjint j4, how- J> ever, being less refracted than 3/, will ap- pear at A\ P at P', &c. ; while iJ, being more refracted than J/, appears at B\ The contraction is sensible only at great zenith distances, where we may assume that AM and PP'Ey small portions of vertical circles drawn through A and P, are sensibly parallel. If then we put S = the true vertical somidiametcr = AMj S^--= the contracted vert, scmid. = -4' J/, aS, = the contracted inclined semid. = J/P', which makes an angle q with the vertical circle, AiS^j = the contraction of the vortical semid. = S — S^ aJS^--zl^ the contraction of the inclined semid. = aS' — JS^ we shall have S^ cos q -- P^E = the difference of the apparent zenith distances of M and P', S^ Z.Z the difference of the app. zen. dist. of J/ and A'. Xow, the difference of the refractions at M and A^ is AA\ and the diflcrence of the refractions at J/ and P' is PP' ; and, since these small difierences are nearly proportional to the difierences of zenith distance, we have S, : *% cos q --: AA' : PP' PP'.--:.S, s -'-^^^? .s. SEMIDIAMETERS. 187 The small triangle PFP^ may be regarded as rectilinear and right-angled at F; whence FP' = PP' X cos q or If we put /Si for S^ in the second member, the resulting value of dSq will never be in error 0".2 for zenith distances less than 85^, and it sufKces to take AS^ = JS, C08« q (253) This formula is sufficiently exact for all purposes to which we shall have occasion to apply it. 134. To find the contraction of the horizontal seniidiameter. — The formula (253) for q = 90° makes the contraction of the hori- zontal semidiameter = 0. This results from our having assumed that the portions of vertical circles drawn through the several points of the limb are parallel, and this assumption de- parts most from the truth in the case of the two ver- tical circles drawn through the extremities of the horizontal diameter. To investigate the error in tliis case, let ZM^ Fig. 23, be the vertical circle drawn through the centre of the body, ZM' that drawn through the extremity of the horizontal semidiameter MM'. In consequence of the refraction, the points M and M' appear at iVand iV'. If we denote the zenith -jf distances of M and N by ^ and z, those of M' and N' by C' a^id 2:', the refraction MN may be expressed as a func- tion either of z or of i^. Art. 107, and we shall have r = k tan z = k' tan C where k and A-' are given by the refraction table with the argu- ments z and ^. The zenith distance of the point M' difters so little from that of M that the values of k and k' will be sensibly the same for both points, and we shall have for the refraction M'N', r' = k tan 2' = A' tan C 188 SEMIDIAMETERS. These two equations give tan z tan C tan if tan C' But if the triangle ZNN' is right-angled at Ny we have tan z cos Z = tan ^ and hence, also, ry tan C cos Z =i tanC' Therefore the triangle ZMM' is also right-angledj and it gives ^ _ tan 5 tan^' tan Z = -T—. — -^ = — . — sm {z + r) sin z in which S = MM' and 5' = NW. Hence tan 5f sin (2: + r) , . 7 — c-/ = — -. — ■ — - = cos r + sin r cot z tan 6' sin 2: ' or, very nearly, %^ = 1 + r sin 1" cot 2 = 1 + ^ sin 1" Hence the contraction of the horizontal semidiameter is ex- pressed by the following formula : S— S' = S' k sin 1" In the zenith, the mean value of log k is 1.76156; at the zenith distance 85^ it is 1.71020. For S' =-- 10', therefore, the contrac- tion found by this formula is 0".27 in the zenith, and 0".24 for 85°. Thus, /or all zenith distahccs less than So^ the contraction of the horizontal semidiameter is very nearly constant and equal to one- fourth of a second, AVhen the body is in the horizon, we have k = root * = 0, and hence *S' — *S'' - 0, which follows also from the sensible parallelism of the vertical circles at the horizon. REDUCTION OF ZENITH DISTANCES. 189 REDUCTION OP OBSERVED ZENITH DISTANCES TO THE CENTRE OF THE EARTH. 135. It is important to observe a proper order in the applica- tion of the several corrections which have been treated of in this chapter. The zenith distance of any point of the heavens observed wHth any instrument is generally affected with the index error and other instnimental errors. These errors will be treated of in the second volume ; here we assume that they have been duly allow^ed for, and we shall call "observed" zenith distance that which would be obtained with a perfect instrument, and shall denote it by z. In all cases the first step in the reduction is to find the refrac- tion r (=ay9^;'^ tan z) with the argument z^ and then z + r is the zenith distance freed from refraction. 1st. In the case oi a fixed star^ C = -2r + r is at once the required geocentric zen. dist. 2d. In the case of the moon., the zenith distance observed is that of the upper or lower limb. If S is the geocentric and S' the augmented semidiameter found by Art. 128, 129, or 130, is the apparent zenith distance of the moon*s centre freed from refraction, and affected only by parallax, and, consequently, it is that which has been denoted by the same symbol in the discus- sion of the parallax. With this, therefore, we compute the parallax in zenith distance, ^' — (^, by Art. 95, and then is the required geocentric zenith distance of the moon's centre. To compute S' by (248), (250), or (251), we must first know C'; but it will suffice to employ in these formulee the approximate value ^' = 2 + r dz /S. AVe can, however, avoid the computation of aS', when extreme precision is not required, by computing the parallax for the zenith distance of the limb. Thus, putting ^' = 2 + r, and 190 REDUCTION OF ZEXITH DISTAXCfiS. compnting C' — C ^^Y ^^- ^'% ^^^ quantity C — C "" (C' "~ C) ^* the geocentric zenith distance of the limb; and therefore, ap- plying the geocentric seniidiameter, ^ ±: S is the required geo- centric zenith distance of tlie moon's centre. This process involves the error of assuming the horizontal parallax for the limb to be the same as that for the moon's centre. It can easily be shown, however, that the error in the result will never amount to 0".2, which in most cases in practice is unimportant. The exact amount will be investigated in the next article. 3d. In the case of the s^m or a planet^ when the limb has been obser\'ed, the process of reduction is, theoretically, the same as for the moon ; but the parallax is so small that the augmentation of the semidiameter is insensible. We therefore take V = z + r ± S and then, computing the parallax by Art. 96, or even by Art. 90, ^ — ^' — (C' ~ C) 5^ ^^^ ^^"^ geocentric zenith distance. If a point has been referred to the sea horizon and the measured altitude is H^ then, D behig the dip of the horizon, A' = II— D is properly the observ^ed altitude, and z = 90® — A' the observed zenith distance, with which we proceed as above. 136. The process above given for reducing the observed zenith distance of the moon's limb to the geocentric zenith distance of the moon's centre, is that which is usually employed; but the whole reduction, exclusive of refraction, may be directly and rigorously computed as follows. Putting C' = c + r = the apparent zenith distance of the moon's limb corrected for refraction, C = the geocentric zenith distance of the moon's centre, then, *S' being the augmented semidiameter, we must substitute ^' ±. *S" for ^' in the formula* for parallax, and, by (101), we have / sin (C' ± aS') =:= sin C — f» sin r cos (9- — 9?') tan y f cos (C' d: S*) — - cos C — /> sin r cos (^p — /) Multiplying the first of these by cos ^', the second by sin ^', and subtracting, we have KEDUCTION OF ZENITH DISTANCES, 191 ±:f Bin JS' = — sin (C — C) + ^- ^^^^ ^--^ sm (C' — - /') cos y in which /= --. By (245) we have also / sin /S' = sin S and hence the rigorous fonnula sin (C - C) = /> sin TT sin (C - r) ^^^ (^ ~ ^0 zp gin ^r cos y for which, however, we may employ with equal accuracy in practice sin (C' — = /> sin r sin (C — r) =F sin S (254) iu which, J. being the moon's azimuth, we have ^ = (^ — ^') cos A If we put (Art 128) A = ~ = 0.272956 a we have sin S=k sin ;:, and (254) may be written as follows: sin (C — C) = |> sin (C' -- r) =F A] sin ;: (255) For convenience in computation, however, it will be better to make the following transformation. Put Hinp = p sin 7t sin (C' — ;') (256) then (254) becomes sin (C' — C) = sin^ qi sin S = Bin (p ::f S) -\- sin p (1 — cos S) =p sin S (1 — cOBp) = sin ( j? qi /S) + 2 sin p sin* J ^' qi 2 sin >S^ sin* ip where the last t\vo terms never amount to 0''.2, and therefore the formula may be considered exact under the form sin (C — C) = sin (p zp S) qz i (p zp S) sin 1" sin p Bin S Since C' — C and pzpS differ by so small a quantity, there will 192 REDUCTION OF ZENITH DISTANCES. be no appreciable error in regarding them as proportional to their sines ; and hence we have C — : =p T /S H= i (/> =H 'S) 8in;> sin 5 (257) the upper signs being nsed for the upper limb and the lower signs for the lower limb. In this formula, p is the parallax computed for the zenith distance of the limb, and the small term i(/> =P S)sin p sin S may be regarded as the connection for the error of assuming the parallax of the limb to be the same as that of the centre. Example. — ^In latitude (p = 38° 59' N., given the observed zenith distance of the moon's lower limb, z = 47° 29' 58", the azimuth A = 33° 0', Barom. 30.25 niches. At. Therm. 65° F., Ext Therm. 64° F., Eq. hor. par. ;: = 59' 10".20 ; find the geocentric zenith distance of the moon's centre : (Table III.) (^ — ^') = 11' 15" » = ^7^ 29'68''.00 log (9 _ ^') = 2.8293 (Table II.) r = 1 2 .27 log cos A 9.9236 C' = 47 31 .27 log }' 2.7529 y = ^ 26 . (Table III.) log p 9.999428 T — y = ^7 21 34 . log sin T 8.235806 log sin (;' — >') 9.800652 log sin /> 8.101886 p= 43' 28". 09 log sin T 8.235806 S = 16 9 .00 (Art. 128) log (0.272956) 9.436003 p ^ S= 59 87 .09 log sin S 7.r,71899 i (p + S)8in p sin S = Al log sin p sin S 5.7739 C — C = 59 37 .20 log {p 4- S) 3.5535 log J 9.6990 C = 46® Sr 23''.07 log Kp-tS) sinp sin S 9.0264 It IS hardly nceessarj- to observe that if the geocentric zenith distance of the centre of the moon or other body is given, the api»areiit zenith distance of the limb affected by parallax and refraction will be deduced by reversing the order of the steps above exjilaincd. If altitudes are given, we may employ altitudes throughout tlie ooniiaitation, i)utting everjwhere 90° — ^, &c. for ^, &c., and making the necessary obvious modilications in the formulae. TIME BY OBSERVATIONS. 193 CHAPTER V. FINDING THE TIME BY ASTRONOMICAL OBSERVATIONS. 137. We have seen, Art. 55, that the local time at any place is readily found when the hour angle of any known heavenly body is given. This hour angle is obtained by observation, but, a direct measure of it being in general impracticable, we must have recourse to observations from which it can be deduced. The observer is supposed to be provided with a clock, chro- nometer, or watch, which is required to show the time, mean or sidereal, either at his own or at some assumed meridian, such as that of Greenwich. The clock correction* is the quantity which must be added alge- braically to the time shown by the clock to obtain the correct time at the meridian for which the clock is regulated. K we put T = the clock time, T' = the true time, A T = the clock correction, we have T' = T + aT or aT = T—T (258) and the clock correction will be positive or negative^ according as the clock is slow or fast. It is generally the immediate object of an observation for time to determine this correction. At the instant of the observation, the time T is noted by the clock, and if this time agrees vnth the time T' computed from the observation, the clock is correct ; otherwise the clock is in error, and its correction is found by the equation ^T= T' — T. The clock rate is the daily or hourly increase of the clock cor- rection. Thus, if * For breTitjr, I shall use dock to denote anjr time-keeper. Vol. L— 13 194 TIME BY OBSERVATIONS. aT^ = the clock correction at a timo T^, we have dT = the clock rate in a unit of time, aT = A 7; + ^r (T — TJ (259) where T — Tq must be expressed in days, hours, &c., according as 5 y is the rate in one day, one hour, &c. When, therefore, the clock correction and rate have been found at a certain instant T^, we can deduce the true time from the clock indication 7" (or "clock face," as it is often called) at any other instant, by the equation T' = T + aT^ + dT^T--- 2;) (260) If the clock correction has been determined at two different times Tq and Tj the rate is inferred by the equation dT=^-^^^^ f261) But these equations are to be used only so long as we can regard the rate as constant. Since such uniformity of rate cannot be assumed for any great length of time, even with the bcvst clocks (although the perform- ance of some of them is really suii)rising), it is proper to make the intcn'al between the observations for time so small that the rate may be taken as constant for that inter\'al. The length of the inten^al will depend upon the character of the clock and the degree of accuracy required. Example. — At noon. May 5, the correction of a mean time clock is — IC" 4?.30 ; at noon. May 12, it is — 16** 13'.50 ; what is the mean time on May 25, when the clock face is 11* IS* 12*.6, supposing the rate to be unifonn ? May 5, corr. = — 16" 47*.30 " 12, '' = — 16_13^50 Rate in 7 days = + BsTsO 5r= + 4.829 Taking, then, as our starting point T^ — May 12, 0*, we have TIME. 195 for the interval to 7^= May 25, 11* 13"» 12^6, T-T^ = IS** 11* 13* 12'.6 = 13*^.467. Hence we have A To = — 16- 13'.50 dT(T-^ To) = + 1 5 .03 £^T=— 15 8.47 T = 11* 13"> 12'.60 T = 10 68 4 .13 But in this example the rate is obtained for one true mean day, while the unit of the interval 13''.467 is a mean day as shown by the clock. The proper interval with which to com- pute the rate in this case is 13'' 10* 58"' 4M3 = 1»'.457 with which we find aT, = — 16-13'.50 dT X 13.457 = + 1 4 .98 - aT=— 16 8.52 T = 11* 13«> 12'.60 T'=10 58 4.08 This repetition will be rendered unnecessary by always giving the rate in a tinit of the clock. Thus, suppose that on June 3, at 4* 11"* 12*.35 by the clock, we have found the correction + 2* 10^.14 ; and on June 4, at 14* 17- 49'.82, we have found the correction + 2* 19*.89 ; the rate in one hour of the clock will be ^r=±^— = + 0'.2858 84.1104 For practical details respecting the care of clocks and other time-keepers, the methods of comparing their indications, &c., see Vol. n. ; sec also Chapter VII., ''Longitude by Chronometer.** I shall here confine myself to the methods of determining their correction by astronomical observation. Those methods, however, which involve details depending upon the peculiar nature of the instrument with which the ob- servation is made, will be treated very briefly in this chapter, and their full discussion will be reserved for Vol. 11. 196 TIME. FIRST METHOD. — BY TRANSITS, 138. At the instant of a star's passage over the meridian, note the time Thy the clock. The star's hour angle at that instant is = 0*, whence the local sidereal time T' is (Art. 55) T' = a = the star's right ascension. If the clock is regulated to the local sidereal time, we have, therefore, aT=o— T But if the clock is regulated to the local mean time, we first con- vert the Bidereal time a into the corresponding mean time T' (Art. 52), and then we have aT= T — T This, then, is in theory the simplest and most direct method possible. It is also practically the most precise when properly carried out with the transit instrument. But, as the transit iii- strumcnt is seldom, if ever, precisely adjusted in the meridian, the clock time 2' of the true meridian transit of a star is itself deduced from the observed time of the transit over the instru- ment by api)lying proi)er corrections, the theory of which will be fullv discussed in Vol. U. It will there be seen, also, that the time may be found from transits over anv vertical circle. SECOND METHOD. — BY EQUAL ALTITUDES. 130. (A.) Equal nlfititdcs nf a fixed star. — The time of the meri- dian transit of a fixed star is the mean between the two times when it is at the same altitude east and west of the meridian: so tbat the observation of these two times is a convenient substi- tute for tbat of the meridian ])assage when a transit instrument is not available. The observation is most frequently made with the sextant and artificial horizon : but any instrument adapted to the measurement of altitudes may be emjiloyed. It is, however, not required tbat the instrument should indicate the true alti- tude ; it is suflicient if the altitude is tlw i^amc at both ob8er\'a- BY EQUAL ALTITUDES. 197 tions. If we use the same instrument, and take care not to change any of its adjustments betw-ecn the two observations, we may generally assume that the same readings of its graduated are represent the same altitude. Small inequalities, however, may still exist, which will be considered hereafter.* The clock correction will be found directly by subtracting the mean of the two clock times of observation from the com- puted time of the star's transit. Example 1. — March 19, 1856; an altitude of Arcturus east of the meridian was noted at 11* 4"^ 5r.5 by a sidereal clock, and the same altitude west of the meridian at 17* 21"* SO'.O; find the clock correction. East 11* 4-51*.5 West 17 21 30.0 Merid. transit by clock = T = 14 13 10 .75 March 19, Arcturus E. A. = o = 14 9 7 .11 Clock correction = ^T =z — 4 3. 64 This is the clock correction at the sidereal time 14* 9"* 7*.ll or at the clock time 14* 13" 10'.75. Example 2. — March 15, 1856, at the Cape of Good Hope, Latitude 33° 56' S., Longitude 1* 13'" 56* E.; equal altitudes of Spica are observed with the sextant as below, the times being noted by a chronometer regulated to mean Greenwich time. The artificial horizon being employed, the altitudes recorded are double altitudes. East. 2 Alt. Spica. West. 10*20- 0*.5 104° 0' 2* 40" 38*. " 20 28. « 10 « 40 10.5 " 20 55. " 20 " 39 42. 10 20 27.83 2 40 10.17 10 20 27.83 Merid. Transit, by Chronom. = T = 12 30 19 .00 The chronometer being regulated to Greenwich time, we must compute the Greenwich mean time of the star's transit at the Cape (Art. 52). We have * For the method of observing equal altitudes with the sextant, see Vol. II., ♦•Sextant." 12 8 41 .92 23 33 5 .87 12 30 86 .55 2 2 .97 12 28 88. .58 12 80 19 .00 198 TIME. Local sidereal time of transit = a = 18* 17"* 87*. 92 Longitude == — 1 18 56 . Greenwich sidereal time = March 15, sid. time of mean noon = Sid. interval from mean noon = Reduction to mean time = Mean Or. time of star's -> _ local transit / Chronometer time of do. = 7* = Chronometer correction = A 7* = — 1 46 .42 140. (B). Equal altitudes of the sun before and after noon. — K the declination of the sun were the same at both observations, the hour angles reckoned from the meridian east and west would be equal when the altitudes were equal, and the mean of the t^'o clock times of observation would be the time by the clock at the instant of apparent noon, and we should find the clock cor- rection as in the case of a fixed star. To find the correction for the change of declination, let 9 = the latitude of tho place of observation, d = the sun's declination at apparent (local) noon, A(J = tho increase of declination from the meridian to the west observation, or the decrease to tho east observation, h = tho sun's true altitude at each observation, T„ = the mean of the clock times A.M. and P.M., aTo = tho correction of this mean to reduce to tho clock time of apparent noon, t = half tho elapsed time between the observations. Then we have t -\- ^Tq=^ tho hour angle at tho A.M. observation reckoned towards the east, t — A 2^^= tho hour angle at the P.M. observation, ^ — Ao = tho declination at the A.M. " a + ArJ = '* " P.M '' and, by the first equation of (14) applied to each observation, sin li = sin <p sin («J — ao) + cos <p cos (d — a*)) cos {t -\- a T^ Bin A = sin ^ sin ((5 + a5) -f cos tp cos {d -f ^^) ^^^ (J — ^ ^») BY EQUAL ALTITUDES. 199 If we substitute sin (^ ± A^) = sin d cos Ad ±: cos d sin ^d cos (d -±2 Ad) = COS 8 COS Ad =p sin d sin Ad cos (< ± A To) = COS t COS A To T sin < sin a T^ and then subtract the first equation from the second, we shall find = 2 sin 9 cos d sin Ad — 2 cos f sin d sin Ad cos t cos a T^ -|- 2 cos f cos d sin t cos Ad sin a T^ whence, by transposing and dividing by the coefficient of sin a T^ ^ tan Ad . tan q> , tan Ad . tan d ^ sm aTI = : ^ H cos a Jl sin ^ tan t This is a rigorous expression of the required correction a 7^, but the change of declination is so small that we may put a5 for its tangent, a 7^ for its sine, and unity for cos aTJ^, without any appreciable error ; and, since a5 is expressed in seconds of arc, we shall obtain aT^ in seconds of time by dividing the second member by 15. We thus find the formula* aT = — ^^- ^^"_? . ^^'^rid " 15 8in< "^ 15tan< ^ ^ The Ephemeris gives the hourly change of d. If we take it for the Greenwich instant corresponding to the local noon, and call it A 'dj and if t is reduced to hours, we have Ad = A'd . ^ and our formula becomes aT.^: — A^d.ftany aM . f tan d pEquationl ^263^ 15 sin t 15 tan t Lfor noonJ To facilitate the computation in practice, we put A = ^~ — — Bz= 15 sin t 15 tan t a=^.A'd.tanf 6 = J?. A'd.tand ) (264) then we have * As first giyen by Gauss, Monatliche Corretpondenz^ Vol. 28. 200 TIME. The correction aT^ is called the equation of equal altitudes. The computation according to the above form is rendered extremely simple by the aid of our Table IV., which gives the values of log A and log J5 with the argument "elapsed time" (=20. Then a and b are computed as above, the algebraic signs of the several factors being duly observed. When the sun is mo\'ing towards the 7iorthy give a'<J the positive sign ; and also when ^ and 3 are north, give them the positive sign ; in the opposite cases they take the negative sign. The signs of A and J5 are given in the table ; A being negative only when t < 12* and B positive when ^ < 6* or > 18*. When we have applied ^T^to the mean of the clock times (or the "middle time"), we have the time as shown by the clock at the instant of the sun's meridian transit Then, computing the time T', whether mean or sidereal, which the clock is required to show at that instant, we have the clock correction, as before, at= r^ t Example. — March 6, 1856, at the U. S. Naval Academy, Lat 38° 59' N., Long. 5* 5"* 57\5 W., the sun was observed at the same altitude, A.M. and P.M., by a chronometer regulated to mean Greenwich time ; the mean of the A.M. times was 1* S" 26'.6, and of the P.M. times 8* 45"" 4r.7 ; find the chronometer cor- rection at noon. Wo have first A.M. Chro. Time = 1* 8- 2(>'.G P.M. '' *' =8 45417 Elapsed time 2< =7 37 15.1 Middle time T; = 4 57 4 .15 From the Ephemeris we find for the local apparent noon of March 5, 1856, ^ :r^ — 5° 46' 22".5 Equation of time = + 11- 35M1 A'a = + 58'MO For the utmost precision, we reduce ^'d to the instant of local BT EQUAL ALTITUDES. 201 noon. With these quantities and ^ = 38® 59', we proceed as follows : Arg. 7» 37- Table IV. log A n9.4804 log B 9.2151 logA'^ 1.7642 logA'^ 1.7642 log tan ^ 9.9081 log tan d n9.0047 log a nl.l527 log 6 n9.9840 a = — 14'.21 b = — 0*.96 Middle Chro. time T^ = 4» 67- 4'.15 ^T^ = a + b = — 15.17 Chro. Time of app. noon T = 4 56 48.98 This quantity is to be compared with the Greenwich time of the local apparent noon, since the chronometer is regulated to Greenwich time. We have Mean local time of app. noon = 0* 11* 35*.ll Longitude =5 5 57 .50 Mean Greenwich time " = T' = 5 17 32.6 1 A r = r' — T = + 20- 43'.63 If the correction of the chronometer to mean local time is required, we have only to omit the application of the longitude. Thus, we should have Chro. time of app. noon = 4* 56* 48'.98 Equation of time = — 11 35 .11 Chro. time of mean noon = 4 45 13 .87 and since at mean noon a chronometer regulated to the local time should give 0* 0** 0*, it is here fccstj and its correction to local time is — 4* 45"* 13'.87. 141. (C.) Efjual altitudes of the sun in the afternoon of one day and the morning of the next folloxcing day ; i.e, hrforc and after midnight, — It is evident that when equal zenith distances are observed in the latitude + ^, their supplement to 180° may be considered as equal zenith distances ob8er\"ed at the antipode in latitude — (p on the same meridian. Hence the formula (263) will give the equation for noon at the antipode by substituting — <f for + ^, that is, by changing the sign of the first term ; but this noon at 202 TIME. the antipode is the same absolute instant as the midnight of the observer, and hence £^T = ^'^ ' ^ ^^^ ^ -t- ^'^-^^^"^ pquation fori ,^^. ° 15 sin ( 15 tan t L midnight. J and this is computed with the aid of the logarithms of A and B in Table IV. precisely as in (264), only changing the sign of A. The sign for this case is given in the table.* 142. To fold the correction for small inequalities in the altitudes. — If from a change in the condition of the atmosphere the re- fraction is different at the two observations, equal apparent alti- tudes will not give equal true altitudes. To find the change ^t in the hour angle t produced by a change aA in the altitude A, we have only to difterentiate the equation sin A =: sin ^ sin d -f cos 9 cos d cos t regarding (p and d as constant ; whence cos h , ^h =z — cos tp cos d sin t . \b^t where aA is in seconds of arc and ^t in seconds of time. If the altitude at the icest observation is the greater by aA, the hour angle is increased by a^ and the middle time is increased by J a/. The correction for the difference of altitudes is therefore — \ a/, and, denoting it by a' 7^y, we have, by the above equation, A' 7;=. ^'^-•-"'*^* (266) 30 cos ff cos ^ sin ^ This correction is to be added algebraically to the middle clock time in any of the cases (A), (13), (C) of the preceding articles. Example. — Suppose that in Example 2, Art. 139, there had been observed at the east observation Barom. 30.30 inches. Therm. 35° F., but at the west observation Barom. 29.5/) inches, Therm. 52° F. AVe have for the altitude 52° 5' or zenith dis- tance 37° 55', bv Table I., the mean refraction 45".4. By Table * For an example and some practical remarks, nee my *' lmprove4l meili(Hl of fin»liii}jj tlie error ami rate of a chronometer by equal altituder*/' Appendix lo the American Kpliemeris for 185C and 1857. BY EQUAL ALTITUDES. 20 o G log aA 0.415 5' log cos h 9.789 66 log 80C 9 0.081 25 log sec d 0.007 51'. log cosec i t 0.270 iog 3 8.523 2 log^'7; 9.085 XIV.A and XiV.B, the corrections for the barometer and ther- mometer are as follows, taking for greater accuracy one-eighth of the corrections for 6' : East Obs. West Obs. Barom. 30.30 + 0".5 Barom. 29.55 — 0".6 Therm. 35°. + 1 .4 Therm. 52°. —0.1 + 1 .9 — .7 The difference of these numbers gives aA = + 2".6 as the excess of the true altitude at the west observation. Hence, by the formula (266), A/i = + 2".G h= 52^ ip = — 33 a = — 10 t = \ elapsed time = 2* 9* a'7;=+ 0M2 When, however, several altitudes have been observed, as in this example, we may obtain this correction from the observa- tions themselves ; for we see that the double altitude of Spica changed 20' = 1200" in about 55*, and hence we have the proportion 1200" : 2".6 = 55* : a' T; which gives aTo = + 0*.12 as before. By taking the change in the double altitude, the fourth tenn is the value of Ja^, or a'TJ,. If this correction be applied, we lind the corrected time of transit = 12* 30"* 19*.12, and consequently the chronometer cor- rection Ar= — 1"* 45*.54. The altitudes may difter from other causes besides a change in the refraction ; for instance, the second observation may be in- termitted by passing clouds, so that the precisely corresponding altitude cannot be taken ; but, rather than lose the whole ob- servation, if we can observe an altitude diftcring but little from the first, we may use it as an equal altitude, and compute the correction for the difterence by the formula (260). 143. Effect of oTors in the latituik^ (Iccllnaiion, and aliltnde upon the time found by equal altitudes. — The time found by e([ual altitudes of a fixed star is wholly independent of errors in the latitude 204 TIME. and declination, since these quantities do not enter into the com- putation. In observations of the sun, an error in the latitude afiects the term a =^ A^'d tan f by differentiating which we lind that an error d(p produces in a the error da = A a'5 . sec^ f . df^ or, putting sin dtp for rf^, da = A ^'d see' ^ sin df In the same manner, we find that an error dd in the declination produces in b the error db = B^'d see' d sin dd In the example of Art. 140, suppose the latitude and declina- tion were each in error V. We have logAr^'d nl.2446 log B^'S 0.9793 logsec'v^ 0.2188 log sec« ^ 0.0044 log sin 1' _6.4637 log sin 1' 6.4687 log da n7.9271 logdb 7.4474 da=— 0'.008 db=-\- 0*.003 If dtp and dd had opposite signs, the whole error in this case would be 0*.008 + 0\003 = O'.Oll. As the obsci-ver can always easily obtain his latitude within 1' and the declination (even when the longitude is somowhat uncertain) within a few seconds, we may regard the method as practically free from the effects of any errors in these quantities. The accuracy of the result will there- fore depend wholly ui)on the accuracy of tlie observations. The accuracy of the observations dci>ends in a measure upon the constancy of the instrument, but chiefly upon the skill of the observer. Each observer may determine the probable error of his observations by discussing them by the method of least squares. An example of such a discussion will be given in the foUowiuic article. The effect of an error in the altitude is given by (266). Since we have, A being the azimuth of the object, cos ^ sin t sm A = cos h BY EQUAL ALTITUDES. 205 the formula may also be written A'7; = ^ 30 cos ^ sin A which will be least when the denominator is greatest, i.e. when A = 90° or 270°, or when the object is near the prime vertical. From this we deduce the practical precept to take the observations whai the object is nearly east or loest. This rule, however, must not be carried so far as to include observations at very low altitudes, where anomalies in the refraction may produce unknown dif- ferences in the altitudes. If the star's declination is very nearly equal to the latitude, it will be in the prime vertical only when quite near to the meridian, and then both observations may be obtained within a brief interval of time ; and this circumstance is favorable to accuracy, inasmuch as the instrument will be less liable to changes in this short time. 144. Probable error of observation. — The error of observation is composed of two errors, one arising from imperfect setting of the index of the sextant, the other from imperfect noting of the time ; but these are inseparable, and can only be discussed as a single error in the observed time. The individual observations are also aftected by any irregularity of graduation of the sextant, but this error does not aftect the mean of a pair of obser\'ations on opposite sides of the meridian ; and therefore the error of observation proper will be shown by comparing the mean of the several pairs with the mean of these means. K, then, the mean of a pair of observed times be called a, the mean of all these means a^,, the probable error of a single pair, supposing all to be of the same weight, is* I(a-a,y Vl{a—a. in which n = the number of pairs, and q = 0.6745 is the factor to reduce mean to probable errors. The probable error of the final mean a^ is r„ = l/n * See Appendix I Leatt Squares. 206 TIME. Example. — At the U. S. Naval Academy, June 18, 1849, the following series of equal altitudes of the sun was observed. Chro. A.M. Chro .P.M. a a — Oo (a-«J^ 0» 43^ 53'. 9*44- ' 3'. 5 5* 13"» 58'.25 -f 0'.12 0.0144 44 19. 43 38. 58.50 + 0.37 .1869 44 45. 43 11.5 58.25 4-0.12 .0144 45 11. 42 46.3 58.65 + 0.52 .2704 45 37. 42 19.7 58.85 + 0.22 .0184 46 1.7 41 53.5 57.60 — 0.58 .2809 46 28.5 41 27. 57 .75 0.38 .1444 46 55. 41 0.5 67.75 0.88 .1444 47 19.7 40 36.5 6" 58.10 — 0.08 2(«-«o)*- .0009 13 58.13 1.0651 » = 9 ii--l=8 r — 0*245 0*.062 A similar discussion of a number of sets of equal altitudes of the sun taken by the same observer gave 0'.23 as the probable error of a single pair for that observer, and consequently the probable error of the result of six observations on each side of the meridian would be only 0'.23 -- 1/ 6 = 0\094. This, liow- ever, expresses only the accidental error of obsaTation^ and does not include the eftcct of changes in the state of the sextant be- tween the morning and afternoon observations. Such changes are not unfrcqncntly produced by the changes of temperature to which it is exposed in obsen'ations of the sun; it is important, therefore, to guard the instrument from tiie sun's rays as much as possible, and to exjjose it only during the few minutes required for each observation. The determination of the time by stars is mostly free from difficulties of this kind, but the observation is not otherwise so accurate as that of the sun, ex- cept in the hands of very skilful observers. THIRD METHOD. — BY A SINGLE ALTITUDE, OR ZENITH DISTANCE, 145. Let the altitude of anv celestial bodv be observed with »' • the sextant or anv altitude instninieut, aud the time noted bv tlie clock. For greater precision, observe several altitudes in ([uick succession, noting the time of each, and take the mean of the altitudes as corre.si)onding to the mean of the times. But BY A SINGLE ALTITUDE. 207 hi taking the mean of several observations in this way, it mnst not be forgotten that we assume that the altitude varies in pro- portion to the time, which is theoretically true only in the exceptional ease where the observer is on the equator and the star's declination is zero. It is, however, practically tnie for an interval of a few minutes when the star is not too near the meridian. The obser\'ations themselves will generally show the limit beyond which it will not be safe to apply this rule. When the observations have been extended beyond this limit, a cor- rection for the unequal change in altitude (i.e. for second diifer- ences) can be applied, which will be treated of below. With the altitude and azimuth instrument we generally ob- tain zenith distances directly. In all cases, however, we may suppose the observation to give the zenith distance. Having then corrected the observation for instrumental errors, for re- fraction, &c.. Arts. 135, 136, let ^ be the resulting true or geo- centric zenith distance. Let (p be the latitude of the place of observation, d the star's declination, / the star's hour angle. The three sides of the spherical triangle formed by the zenith, the pole, and the star may be denoted by a = 90° — ^, 6 = ^, e = 90° — 5, and the angle at the pole by 13 = ^, and hence. Art. 22, we deduce ,inj,_ /( Bi»K: + (y -3)] sin K:-(y -«>)] ) (207) \ \ cos yj cos d I which gives / by a very simple logarithmic computation. From / we deduce, by Art. 55, the local time, which compared with the ob3er\'ed clock time gives the clock correction required. It is to be obsei^ved that the double sign belonging to the radical in (267) gives t\vo values of sin J /, the positive corre- sponding to a west and the negative to an east hour angle; since any given zenith distance may be observed on either side of the meridian. To distinguish the true solution, the observer must of course note on which side of the meridian he has observed. If the object ob8er\x»d is the sun, the moon, or a planet, its declination is to be taken from the Ephemeris, for the time of the obser\'ation (referred to the meridian of the Ephemeris); but, as this time is itself to be found from the observation, we must at first assume an approximate value of it, with which an approxi- mate declination is found. With this declinatibn a first compu- 208 TIME. tation by the formula gives an approximate value of t, and hence a more accurate value of the time, and a new value of the decli- nation, with which a second computation by the formula gives a still more accurate value of L Thus it appears that the solution of our problem is really indirect, and theoretically involves an infinite series of successive approximations; in practice, how- ever, the observer generally possesses a sufficiently precise value of his clock correction for the purpose of taking out the declina- tion of the sun or planets. The moon is never employed for determinhig the local time except at sea, and when no other object is available.* Example. — At the U. S. Naval Academy, in Latitude <p = 38° 68' 53" N., Longitude 5* 5- 57'.5 W., December 9, 1851, the fol- lowing double altitudes of the sun west of the meridian were observed with a sextant and artificial horizon, the times being noted by a Greenwich mean time chronometer: Chronometer. 2^1 7* 35- 14'.5 SS*' 30' Barom. 30.28 inches. 35 55. « 20 Att. Therm. 55** F. 36 35.5 " 10 Ext. Therm. 50^ F. 37 15.5 " Index correction of the 37 55. 32 50 soxtaut — 1' 10" Means 7 36 35.1 33 10 The approximate correction of the chronometer was assumed to be + 9"* 40*. Find its true correction. With the assumed chronometer correction we obtain the aj> proximate Greenwich time = 7* 46"* 15% with which we take from the Ephemeris d=.-. — 22° 50' 27" Sun's semidiametcr S = 16' 17" Eq. of time :^ — 7" 25'.80 " hor. parallax r = 8".7 "We have then ♦ But the moon's altitude and the hour nngle deduced from it may be used in finding the observer's longitude, as will be shown in the Chapter on Longitude. t The -ymbol Q is used for "observed altitude of the sun's lower limb/* and 2Q for the double altitude from the artificial horizon. In a similar manner we um Cy» X 'j>' BY A SINGLE ALTITUDE. 209 Observed 2 © =83^ 10' 0" Index corr. = — 1 10 33 8 50 App. altitude = 16 34 25 ^r = 73 25 35 (Table II.) r = + 3 15 7c Gin z = p =2 — 8 /S= — 16 17 C = 73 12 25 The computation by (267) is then as follows : sr= 38^58' 53" log sec s«^ 0.109383 ^ = — 22 50 27 log sec d 0.035464 ^ — ^ = 61 49 20 log sin } sum 9.965661 C = 73 12 25 log sin } diff. 8.996455 i sum = 67 30 52 .5 19106963 J diflf. = 5 41 32 .5 log sin } f 9.553482 J f = 20« 57' 25".6 Apparent time = ^ = 2* 47"» 39'.4 Eq. of time = —7 25.8 Local mean time = 2 40 13 .6 Longitude =5 5 57 .5 True Gr. Time =T'= 7 46 11.1 T= 7 36 35.1 A!r=+ 9 36.0 agreeing so nearly with the assumed correction that a repetition of the computation is unnecessary. 146. If it is preferred to use the altitude instead of the zenith distance, put the true altitude h = 90° — ^, and the polar distance of the star P = 90° — d, then we have, in (267), sini[C— (f — ^)]=8inJ(90° — A — ^ + 90°— P) = cosJ(A+^ + P) siniCC+f' — ^]=sinj(90°— A + sp — 90^ + P) = 8m J(^ + P^A) If then we put the formula becomes sin Vol. L— 14 . ,^ //C08 5 sin (5 -- A)\ \\ cosy>sinP / ^ ' 210 TIME. In this form we may always take P = tlie distance from the ele- vated pole, and regard the latitude as always positive, and then no attention to the algebraic signs of the quantities in the second member is required. Thus, in the preceding example, we should proceed as follows : . App.alt.= 16^ 34' 25" r— ;>= — 3 7 S= 16 17 A = 16 47 35 ^= SS 5S 53 log sec 0.109383 P = 112 50 27 log cosec 0.035464 25 = 168 36 55 8= 84 18 27 .5 log cos 8.996455 5 — A = 67 30 52 .5 log sin 9.965661 19.106963 and the computation is finished as in the preceding article. 147. If we aim at the greatest degree of precision which the logarithmic tables can afford, we should find the angle J/ by its tangent, since the logarithms of the tangent always vary more rapidly than those of the other functions. For this purpose wo deduce tanit = Jl Bi!L(^^^)^^n.(.-a) ^ } (269) ' \ cos S COS (s — C) / or, if the altitude is used, s = 1 (A + y + -P) ) ^ \ sin (s — ^) cos {s — P) / 148. If a number of observations of the same star at the same place arc to be individually computed, it will be most readily done by the fundamental equation cos C — Bin <p sin d cos t = cos ^ COB d BY A SINGLE ALTITUDE. 211 for the logarithms of sin y> sin S and cos <p cos 5 will be constant, and for eacli observation we shall only have to take from the trigonometric table the log. of cos ^ ; the logarithm of the nume- rator ^\l]\ then be found by the aid of Zecii's Addition or Sub- traction Table, which is included in IIulsse*s edition of Vega's Tables. The addition or the subtraction table vnll be used ac- cording as sin y> sin d is positive or negative. 149. JEffcci of errors in the data upon the time computed from an altitude. — We have from the dilterential equation (51), Art. 35, multiplymg dt by 15 to reduce it to seconds of arc, sin q cos d (15 dt) = d^ — cos A d<p -j- cos q dd wliere e/^, dip, d8, may denote small errors of ^, y>, d, and dt the corresponding error of t;Ais the star's azimuth, q the parallactic angle, or angle at the star. If the zenith distance alone is erroneous, we have, by putting df = 0, and dd = 0, Udt= "^^ ^^ sin q cos d cos ^ sin A from which it follows that a given error in the zenith distance will have the least eftect upon the computed time when the azimuth is 90^ or 270° ; that is, when the star is on the prime vertical; for we then have m\A = ± 1, and the denominator of this expression obtains its maximum numerical value. Also, since cos y> is a maximum for f> = 0, it follows that observa- tions of zenith distances for determining the time give the most accurate results when the place is on the equator. On the other hand, the least favorable position of the star is when it is on the meridian, and the least favorable position of the observer is at the pole. By putting d!^ = 0yd8 = 0, sin q cos 8 = cos y> sin A we have ibdt=- ^'^ cos ip tan A by which we see that an error in the latitude also produces the least eftect when the star is on the prime vertical, or when the obser^-er is on the equator. Indeed, when the star is exactly in 212 TIME. tlic prime vertical, a small error in ip has no appreciable effect : since, then, tau^l =^ ac, and hence when the latitude is uncertain, we may still obtain good results by obser\'ing only stars near the prime vertical. By putting rf^ = 0, d(p = 0, we have cos J tan (I which shows that the error in the declination of a given star produces tlie least effect when the star is on the prime vertical ;* and of different stars the most eligible is that which is nearest to the equator. As verj' great zenith distances (greater than 80°) are, if jws- sible, to be avoided on account of the uncertainty in the refraction, the observer will often be obliged, especially in high latitudes, to take his obsen'ations at some distance from the prime vertical, in which case snuill erroi's of zenith distance, latitude, or declina- tion may have an important effect upon the computed chxh ror- revtion. Nevertheless, constant errors in these quantities will have no sensible effect upon the rate of the clock deduced from zenith distances of the same star on different davs, if the star is observed at the same or nearly the same azimuth, on the same side of the meridian ; for all the clock corrections will be in- creased or diminished by the same quantities, so that their differences, and consequently the rate, will be the same as if these errors did not exist. The errors of eccentricitv and graduation of the instrument are among the constant errors which niav thus be eliminated. ft. But if the same star is observed both east and west of the meridian, and at the same distance from it, sin .4 or tan^l, and tan 7, will be positive at one observation and negative at the other, and, having the same numerical value, constant errors dif, (ft), and (l;^ will give the same numerical value of (ft with op[»ositc signs. Hence, while one of the deduced clock correc- tions will be too great, the other will be too small, and their mean will be the true correction at the time of the star's transit * From the equation sin g = — sin Ay it follows that sin ^ is a maximum cos (S (for constant values of o and 6) when sin .1 ::= 1, and tan ^ is a maximam in the enme case. CORRECTION FOR SECOND DIFFERENCES. 213 over the meridian. Ilenee, it follows again, as in Art. 143, that small errors in the latitude and declination have no sensible cflect upon the time computed from equal altitudes. 150. To find the change of zenith distance of a star in a given in- terval of time^ having regard to second differences. The formula dZ = COS ^ sin A dt is strictly true only when rfj and dt are infinitesimals. But the complete expression of the finite difference a^ in terms of the finite diflference ^t involves the square and higher powers of ^t. Let ^ be expressed as a function of t of the form C=ft then, to find any zenith distance ^ + ^C corresponding to the hour angle t + a^, we have, by Taylor's Theorem, c + ^:=f(t + An =ft + -^'^t+—^- h--- ^ dt dt^ 2 or, taking only second differences, ^ d: ^ , dK A<' AC = A^ H dt dt' 2 We have already found d: = cos f> sin A dt which gives, since A varies with t, but <p is constant, dK , dA — = cos cos A • — dt* dt But from the second of equations (51) we have, since dd and dtp are here zero, dA cos q cos d cos q sin A dt sin C sin t whence d*^ cos ^ sin A cos A cos g dt* smt 214 TIME. and the expression for a^ becomes . - ^ , cos cp sin^ cosul cosflf ^t^ AC = cos ^BmA.^t-\ ^ sin t 2 Since a^ and a^ are here supposed to be expressed in parts of the radius, if we wish to express them in seconds of arc and of time respectively, we must substitute for them a^ sin V and 15 A< shi 1", and the formula becomes • A ^ir ^N , COR v» sin il cos -4 cos flf flSAO'sinl" ,^.,, AC = cos ^ sin A (15 a^ H • ^^ (271) sin t 2 But in so small a term as the last we may put (15A0'sin 1" 2sin» jAf 2 sin 1" the value of which is given in our Table V., and its logarithm in Table VI. ; so that if we put also . , , cos A cos a a = cos ^ sm-4, A* = ^ sin t we shall have aC = 15aA^ + akm (272) 151. A number of zenith disianees being observed at given o/ocJt times, to correct the mean of the zenith distances or of the clock time^ for second differences, — The first term of the above value of a^ varioH in j^roportiou to a^, but the second term varies in projwr- tion to a/^; and hence, when the interval is sufficiently great to render this second term sensible, equal intervals of time corre- spond to unequal difterences of zenith distance, and vice versa: in otlier words, we shall have second diflerences either of the zenith distance or of the time. Two methods of correction present themselves. 1st. Reduction of the mean of the zenith distances to the mean of the times. — Let 7'„ T^, T^, &c. be the observed clock times ; ^^, ^^ ^j, &c. the corresponding obsei'ved zenith distances; 7* the mean ot the times; i^o ^^^^ mean of the zenith distances; ^ the zenith distance corresponding to 71 The change Ci ~ C eorresjwnds to the interval 7; — 7\ Ca — C ^<^ ^2 — ^\ ^^' ; »« that if we put 7;- T=T,, T,- r = r.,&c. CORRECTION FOR SECOND DIFFERENCES. 215 we have, by (272), Ci — C = 15 a Tj -|- aknii C, — C = 15 a T, + akm^ C, — C = 15 a T, + akm^ &c. &c. m which Tiu = — : — — — % rru = — : — —-5, &c., are found by Tab. V. ^ sin 1" ' ^ sin 1" ' ' -^ with the arguments Tj, r„ &c. The mean of these equations, observing that ■^1 + ^9 + T. + Ac. = gives C=C g/; ^ + ^, + Wl, + &C " n in which n = the number of observations. Or, denoting the mean of the values of m from the table by ?w^„ that is, putting wi, -|- m, + m, + &c. • n we have C = Co - akm, (278) 2rf. Reduction of the mean of the times to the mean of tlie zenith distances. — Let T^ be the clock time corresponding to the mean of the zenith distances, then Za — C i® ^^ change of zenith dis- tance in the interval T^ — jT, and, since this interval is very small, we shall have sensibly 15a (j; — T) = C. — C = akm^ whence T,= T + -r^km, (274) We have, then, only to compute the true time 7^/ from the mean of the zenith distances in the usual manner, and the clock cor- rection will then be found, as in other cases, by the formula To compute A-, we must either first find q and A^ or, which is preferable, express it by the known quantities. We have cos gr cos^ = cos t — sin jr sin A cos C sin'^ == cos t cos ip cos ^ cos C sin'C 216 TIME. whence m mil X a 1 ^^^^ COS f> COS d ,o^cv r, = r + J5 m, cot t - Jy ^« — . ^/ ^ (275) sin C tan C in which we employ for ^ and < the mean zenith distance and the computed hour angle. This mode of correction is evidently more simple and direct than the first. Example.— In St. Louis, Lat. 38° 38' 15" K, Long. 6* 1" 7* W., tne following double altitudes of the sun w^ere observed with a Pistor and Martin prismatic sextant, the index correction of which was + 20". The assumed correction of the chronometer to mean local time was + 2*^ 12*. Barom. 30.25 inches, Att Therm. 80°, Ext. Therm. 81°. St. Louis, June 24, 1861. 2^ 1250 16' 10". 125 49 10 12G 23 12G 41 40 127 32 80 127 67 45 128 22 128 61 50 129 8 35 129 38 Mean 127 33 28 -]- 20 127 33 48 Obs'd 63 4G 54 (*)r — 27 .2 P — -1- 3 .7 S = -1- 15 40 .3 K- 64 2 16 .8 Co = 25 57 43 .2 ^ - 38 38 15 . (5 = 23 23 49 .3 t - 24° 43' 48".4 — . 1*38«55'.23 App. time — 22 21 4.77 Eq. of time ^^ 4- 2 18.17 7" - ^0 - 22 23 22.94 Chronom. r m 22* 14*" '80«.5 6" .42. 88^.14 16 7.6 6 6 50 .73 17 46.0 8 26 28 .14 18 89.6 2 88 12 .76 21 6.6 6 .02 22 22. 1 10 2 .67 23 33.6 2 21 10.84 25 1.2 8 40 28 .60 25 51.3 4 89 42 .46 27 8.5 6 61 67 .19 22 21 12.15 m lo — 82 .66 Correction for 1 ^ «,. second diff. /JI L1 ^^«^o 1-^139 T --22 21 10.48 ^^* ^ 8.8239 ^/ __ 22 03 00 94 ^^K ^^^ ^ «0.3367 AT-:- + 2 12.46 4*. 73 wO.6745 log ^ m, 0.3878 log sin t n9.6215 log cos ^ 9.8927 log cos d 9.9627 log cosec :^ 0.8688 log cot Z^ 0.8125 — 8*.06 fiO.4a60 — 1.67 *Thc refraction should here be the mean of the refractions computed for the CORRECTION FOR SECOND DIFFERENCES. 217 The correction for second difterences is particularly useful in reducing series of altitudes observed with the repeating circle ;* for with this instrument we do not obtain the several altitudes, but only their mean. (See Vol. 11.) Wlien the several altitudes are knowni, we can avoid the correction by computing each observ^ation, or by dividing the whole series into groups of such extent that ^\'ithin the limits of each the second difterences will be insensible, and computing the time from the mean of each group. FOURTH METHOD. — BY THE DISAPPEARANCE OF A STAR BEHIND A TERRESTRIAL OBJECT. 152. The rate of the clock may be found by this method with considerable accuracy without the aid of astronomical instru- ments. The terrestrial object should have a sharply defined vertical edge, behind which the disaj^i^earance is to be obser\'ed, and the position of the eye of the observer should be j^recisely the same at all the observations. If the star's right ascension and declination are constant, the difterence between the sidereal clock times 7^ and T^ of two disappearances is the rate dTm the interval, or but if the right ascension a has increased in the interval by Aa, then the rate is dT=T^— T^ + Aa To find the correction for a small change of declination = a5, several altitudes or zenith distances, but for small zenith distances the difference will be insensible. At great zenith distances we should compute the several refrac- tions, but under 80^ we may take the refraction r for the mean apparent zenith distance z^, and correct it as follows : Take the difference between z^ and each z, and the mean m^ of the values of ^_ 28inM(^-^o) sin \" from Table V. (converting the argument z — z^ into time) ; then the mean of the refractions will be found by the formula r^ = r -j- 2»Iq sin r sec* i^ The difference z — z^ should not much exceed 1°. * This method was frequently practised in the geodetic survey of France. See Nouvelle Deseriplion GSomSlrique de la France (Puisbamt), Vol. I. p. 96. 218 TIME. we have, by the second equation of (51), since the azimuth is here constant as well as the latitude, so that dA = and df = 0, a5 tan a 15 cos d and hence the rate in the interval will be iT=T,-T, + ^a-^^^^ (276) 15 cos d The angle q will be found with sufficient precision from an approximate value of i by (19) or (20). If we know the absolute azimuth of the object, we can find the hour angle by Art. 12, and hence also the clock correction. TIME OP RISING AND SETTING OF THE STARS. 153. To fold the time of true rising or setting, — that is, tlie instant when the star is in the true horizon, — ^we have only to compute the hour angle by the formula (28) cos t =z — tan <p tan d and then deduce the local time by Art. 55. 154. To find the time of apparent rising or setting, — that is, the instant when the star appears on the horizon of the observer, — we must allow for the horizontal refraction. Denoting this refraction by r^, the true zenith distance of the star at the time of apparent rising or setting is 90° + r^, and, employing this value for ^, we compute the hour angle by (2G7). Since the altitude A = 90° — (J^, we have in this case h = — r^ with which we can compute the hour angle by the formula (268). In common life, bv the time of sunrise or sunset is meant the instant when the sun's upper limb appears in the horizon. The true zenith distance of the centre is, then, ^ = 90° -\- r^ — r -•- S (where z - the horizontal parallax and *V = the semidiameter), with which we compute the hour angle as before. The same form is to be used for the moon. TIME OF THE BEfllXXIXG AND ENDING OF TWILIGHT. 155. Twilight begins in the morning or ends in the evening when the sun is 18° below the horizon, and consequently the AT SEA. 219 zenith distance is then ^ = 90° + 18°, or A = — 18°, with which we can find the hour angle by (267) or (268). NoTK. — Methods of finding at once both the time and the latitude from observed altitudes will be treated of under Latitude, in the next chapter. FINDING THE TIME AT SEA. First Method. — By a Single Aliiiudc. 156. This is the most common method among navigators, as altitudes from the sea horizon are observed with the greatest facility with the sextant. Denoting the observed altitude cor- rected for the index error of the sextant by H^ the dip of the horizon by X), we have the apparent altitude h' = H — D; then, taking the refraction r for the argument A', the true altitude of a star is A = A' — r. A i:)lanet is observed by bringing the esti- mated centre of its reflected image upon the horizon, so that no correction for the semidiameter is employed; the jjarallax is com- puted by the simple formula (;r being the horizontal parallax) p = 7: cos A' and hence for a planet h = h' — r -{- IT cos A' The moon and sun are observed by bringing the reflected image of either the upper or the lower limb to touch the horizon. As very great precision is neither possible nor necessary in these observ^ations, the compression of the earth is neglected, and the parallax is computed by the formula p z= n cos (A' — r) and then, S being the semidiameter, A = A' — r + TT cos (A' — r) ± S In nautical works, the whole correction of the moon's altitude for parallax and refraction = ;: cos (A' — r) — r is given in a table with the arguments apparent altitude (A') and horizontal jiarallax (r). In the construction of this table the mean refraction is used, but the corrections for the barometer and thermometer are given in a very simple table, although they are not usually of suflicient importance to be regarded in correcting altitudes of the moon which are taken to determine the local time. 220 TIME. The hour angle is usually found by (268). It is important at sea, where the latitude is always in some degree uncertain, to iind the time by altitudes near tlie prime vertical, where the error of latitude has little or no eftect (Art. 149). 157. The instant when the sun's limb touches the sea horizon may be observed, instead of measuring an altitude with tlie sex- tant. In this case the refraction should be taken for tlie zenith distance 90° + Z), but, on account of the uncertainty in the liori- zontal refraetion, great precision is not to be expected, and the mean horizontal refraction i\ may be used. We then have l^ == QO"" + D + r^^— t: ± Sy with which we proceed by (267). In so rude a method, t: may be neglected, and we may take 16' as the mean value of Sj 36' as the value of r^,, 4' as tlie average value of/) from the deck of most vessels; then for the lower limb we have C = 90° 56', and for the upper limb r = 90° 24'. If both limbs have been observed and the mean of the times is taken, the corresponding hour angle will be found by taking C = 90° 40'. Second Method. — By Equal Altitudes. 158. The method of equal altitudes as explained in Arts. 139 and 140 may be applied at sea by introducing a correction for the ship's change of place between the two observations. If, however, the ship sails due east or west between the obsen'a- tions, and thus without changing her latitude, no correction tor her change of place is necessary, for the middle time will evi- dently correspond to the instant of transit of the star over the middle meridian between the two meridians on which the equal altitudes are observed. But, if the ship changes her latitude, let Af = the increase of latitude at the second observation; then (Art. 149) the eftect upon the second hour angle is Af r=: — 15 cos <p tan A which is the correction subtractive from the second obBorved time to reduce it to that which would have been observed if the AT SEA. 221 ship had not changed her latitude or had run upon a parallel. Hence Ja< is to be subtracted from the mean of the chrono- meter times to obtain the chronometer time of the star's transit over the middle meridian. In this formula we must observe the sign of tan A, It will be more convenient in practice to disregard the signs, and to apply the numerical value of the correction to the middle time according to the following simple rule : — add the correction when the ship has receded from the sun; subtract it when the ship has approached the sun. The azimuth may be found by the formula . sin t cos d sm A = cos h in which for t we take one-half the elapsed time. The sun being the only object which is employed in this way, we should also apply the equation of equal altitudes, Art. 140; but, as the greatest change of the sun's declination in one hour is about 1', and the change of the ship's latitude is generally much greater, the equation is commonly neglected as relatively unimportant in a method which at sea is necessarily but ap- proximate. But, if required, the equation may be computed and applied precisely as if the ship had been at rest. Example. — At sea, March 20, 1856, the latitude at noon being 39° N., the same altitude was obser\^ed A.M. and P.M. as fol- lows, by a chronometer regulated to mean Greenwich time : Obsd-X^ 30° 0' A.M. Chro. time = 11* 39-» 33 Index corr. — 2 P.M. *' " = 6 20 17 Dip — 4 Elapsed time = 2^ = 6 40 44 Refraction — 2 Middle time = 2 59 55 Semidiam. + 16 Chron. correction = — 2 12 A = 30 8 Green, time of) 9 57 4^ noon J The ship changed her latitude between the two observations by A^ = — 20' = — 1200". For the Greenwich date March 20, 2* 58'", the Ephemeris gives 5 = + 0° 4', and we have t = 8* 20« 22- = 50° 5' 30'', ip = 39° 0'. Hence 222 TIME. log sin t 9.8848 log j'^ 8.5229 log cos d 0.0000 log A^ 3.0792 log sec h 0.0631 log sec <p 0.1095 log sin A 9.9479 log cot A 9.7165 log26'.8 1.4281 The ship has approached the sun, and hence 26*.8 must be sub- tracted from the middle time. K we wish to apply the equation of equal altitudes, we have further from the Ephemeris £^'d = + 59", and hence, by Art 140, log A n9.4628 log 5 9.2698 logA'^ 1.7709 logA'd 1.7709 log tan *p 9.9084 log tan d 7.0658 a = — 13v9 log a nl.l421 6 = + O-.O log h 8.1065 Hence we have Chro. middle time = 2» 59"» 55*. Corr. for change of lat. = — 26 .8 Equation of eq. alts. = — 13 .9 Chro. time app. noon = 2 59 14 .3 At sea, instead of using the observation to find the chrono- meter correction, we use it to determine the ship's longitude (as will be fully shown hereafter) ; and therefore, to carrj' the opera- tion out to the end, we shall have Chro. time app. noon = 2* 59"* 14* Corr. of ehronom. = — 2 12 Green, mean time noon = 2 57 2 Equation of time = — 7 48 Greenwich app. time at the local noon = 2 49 14 which is the longitude of the middle meridian, or the longitude of the ship at noon. 159. In low latitudes (as within the tropics) observations for the time may be taken when the sun is very near the meridian, for the condition that the sun should be near the prime vertical may then be satisfied within a few minutes of noon ; and in ease the ship's latitude is exactly equal to the declination, it will be satisfied only when the sun is on the meridian in the zenith. In such cases the two ecjual altitudes may be obser\'ed within a few minutes of each other, and all corrections, whether for change of latitude or change of declination, may be disregarded. MEBIDIAN ALTITUDES. 223 CHAPTER VI. FINDING THE LATITUDE BY ASTRONOMICAL OBSERVATIONS. 160. By the definition, Art. 7, the latitude of a place on the surface of the earth is the declination of the zenith. It was also shown in Art. 8 to be equal to the altitude of the north pole above tlie horizon of the place. In adopting the latter definition, it is to be remembered that a depression below the horizon is a negative altitude, and that south latitude is negative. The south latitude of a place, considered numerically, or without regard to its algebraic sign, is equal to the elevation of the south pole. It is to be remembered, also, that the latitude thus defined is not an angle at the centre of the earth measured by an arc of the meridian, as it would be if the earth were a sphere ; but it is the angle which the vertical line at the place makes with the plane of the equator. Art. 81. We have seen. Art. 86, that there are abnormal deviations of the plumb line, which make it necessary to distinguish between the geodetic and the astronomical latitude. We shall here treat ex- clusively of the methods of determining the astronomical lati- tude; for this depends only upon the actual position of the plumb line, and is merely the declination of that point of the heavens towards which the plumb line is directed. FIRST METHOD. — BY MERIDIAN ALTITUDES OR ZENITH DISTANCES. 161. Let the altitude or zenith distance of a star of known declination be observed at the instant when it is on the meridian. Deduce the true geocentric zenith distance ^, and let d be the geocentric declination, (p the astronomical latitude. Let the celestial sphere be projected on the plane of the meridian, and let ZNZ\ Fig. 24, be the celestial meridian: the centre of the sphere coincident with that of the earth ; PCP' the axis of the sphere; P the north pole; and ECQ the projection 224 LATITUDE. of the plane of the equinoctial. Let CZ be parallel to the vertical line of the obsen-er; then the point ^of the celestial sphere, being the vanishing point of all lines parallel to CZ^ is the astronomical zenith of the obserx'er, and ZE=^ the astro- nomical latitude = if. If, tfien, A is the position of the star on the meridian, north \y of the equator but south of the zenith, we have ZA = ^, AE = 5, and hence f = a + C (277) This equation may be treated as entirely general by attending to the signs of d and ^. Since in deducing it we supposed the star to be north of the equator, it holds for the case where it is south by giving the declination in that case the negative sign, according to the established practice; and, since we supposed the star to be south of the zenith, the equation will hold for the case where it is north of the zenith by giving ^ in that case the negative sign. If the star is so far north of the zenith as to be below the pole, or at its lower culmination, the equation will still hold, pro\'ided we still understand by o the star's distance north of the equator, measured from E through the zenith and ekrattd poU\ or the arc EA\ This arc is the supplement of the declination ; and we may here remark that, in general, any formula deduced for the case of a star above the pole will apply to the case where it is below the pole by emi»loying the 8upi»lenu*nt of the declination instead of the declination itself; that is, by reckoning the declination ovtr the jyolc. The ease of a star beU)w the pole is, however, usually con- sidered under the following simple form. Put P = PA' = the star's polar distance, /* = yA' = ** true altitude, then ^ =P-{-h (278) in whii'h for south latitude P must be the star's south polar dis- tance, and the sum of P and h is only the numerical value of f . The declination is to be found for the instant of the meridian transit bv Art. GO or 02. In the obser\'atory, instruments are employed which give MERIBUN ALTITUDES. 225 directly the zenith distance, or its supplement, the nadir distance. "With a meridian circle perfectly adjusted in the meridian^ the instant of transit would be known without reference to the clock, and the observation would be made at the instant the star passed the middle thread of the reticule ; but when the in- strument is not exactly in the meridian, or when the observation is not made on the middle thread, the observed zenith distance must be reduced to the meridian, for which see Vol. 11., Meridian Circle. "With the sextant or other portable instruments the meridian altitude of a fixed star may be distinguished as the greatest altitude, and no reference to the time is necessary. But, as the sun, moon, and planets constantly change their declination, their greatest altitudes may be reached either before or after the meridian passage ;* and in order to observe a strictly meridian altitude the clock time of transit must be previously computed and the altitude observed at that time. Example 1. — On March 1, 1856, in Long. 10* 5* 32* E., suppose the apparent meridian altitude of the sun's upper limb, north of the zenith, is 63° 49' 50", Barom. 30. in., Ext. Therm. 50° ; what is the latitude ? App. zen. diat. Q_ — 26» W 10". r — + 28 .7 P — — 8 .8 8 = + 16 10 .3 C =■ - 26 26 45 .2 d = — 7 33 5 .8 ^ = — 33 59 51 .0 Example 2. — July 20, 1856, suppose that at a certain place the true zenith distances of a Aquilce south of the zenith, and a Cephei north of the zenith, have been obtained as follows : a Aquilm a Cephei C = + 26^ 34' 27".5 C = — 26^ 54' 28".3 d = + 8 29 22 7 ^ = + 6 1 58 21 .1 ^ = + 35 3 50 .2 ^ = + 35 3 52 .8 The mean latitude obtained by the two stars is, therefore, f = + 35° 3' 51".5. In this example, the stars being at nearly * See Art. 172 for the method of finding the time of the 8un*8 greatest altitude, which may also be used for the moon or a planet. Vol. I.— 16 the same zenith distance, but on opposite sides of the zenith, my constant though unknown error oj" the iiiefrument, peculiar to that zenith diBfance, is eliminated in tJiking the meiiD. Thus, if tlie zenith distance in both caees had been 10" greater, we should have found from a Aqtiila: f = 35* 4' 0".2, but from a CepUi p = 35° 3' 42".8, but the mean would still be ^ = 35* 3' 51".5. It is evident, also, that errors in the refraction, whether due to the tables or to constant errors of the barometer and tliemio- meter, or to anj peculiar state of the air common to the t^vo observations, are nearly or quite eliminated by thus combining a pair of stars the mean of whose declinations is nearly equal to the declination of the zenith. The advantages of such a com- bination do not end here. If we select the two stars so that the diilerence of their zenith distances is so small that it may be measured with a micrometer attached to a telescope which is eo mounted that it may be successively directed upon the two stars without disturbing the angle which it makes with the vertical line, we can dispense altogether with a graduated circle, or, at least, the result obtained will be altogether independent of its indicatiouB. For, let J and f ' be the zenith distances, i and i' the declinations of the two stars, the second of which is north of the zenith; then, if J' denotes only the numerical value of the zenith distance, we have the mean of which is f = U3 + + i(c-r') (2") 60 that the result depends only upon the ^ven declinations and the observed difference of zenith distance which is measured with the micrometer. Such is the simple principle of the method first introduced by Captain Talcott, and now extensively nsed in thin country. To give it full effect, the instrument formerly known as the Zenith Telescope in England baa received several imiiortaot modifications from our Coast Survey. It will he fully treated of, in its present improved form, in Vol. II., where also will be found a discussion of Talcott's method in all its details, 162. Meridian nlliludes of a eircnrnpofar stnr observed both abort and btlotc the pole. — Every star whose distance from the elevated MERIDIAN ALTITUDES. 227 pole is less than the latitude may be observed at both its upper and lower culminations. If we put h = the true altitude at the upper culmination, Aj= " '' *' lower " p = the star's polar distance at the upper culmination, p^ = " " " " lower « we have, evidently, the mean of which is 9 = H^ + K) + HPi-P) (280) whence it appears that by this method the absolute values of p and j?! are not required, but only their difference p^ — p. The change of a star's declination by precession and nutation is so small in 12* as usually to be neglected, but for extreme precision ought to be allowed for. This method, then, is free from any error in the declination of the star, and is, therefore, employed in all fixed observatories. Example. — ^With the meridian circle of the !N*aval Academy the upper i^nd lower transits of Polaris were observed in 1853 Sept. 15 and 16, and the altitudes deduced were as below: Upper Transit. Lower Transit. Sept. 16, App. alt. 40O28'26".42 Sept. 16, ST® 81' 89". 76 Barom. 80.005 ^ Barom. 80. 146 ^ Att. Therm. 66<>.2 I Ref. 16 .84 Att. Therm. 75«» [ Ref. 1 12 .46 Ext. " 68 .8 3 Ext. " 74 .63 A = 40 27 19 .08 h^ = 87 80 27 .81 p = 1 28 26 .04 p^= 1 28 26 .86 = 88 68 68 .04 ^ = 88 58 63 .17 *' " 63 .04 Mean ^ = 88 68 68 .11 In order to compare the results, each observation is carried out separately. By (280) we should have i{h + A,) = 38^ 58' 53".20 i(j>i'-P)= — 09 f = 38 58 53 .11 This method is still subject to the whole error in the refhiction, 228 LATITUDE. which, however, in the present state of the tables, will usually be rery small. If the latitude is greater than 45°, and the star's declination less than 45°, the upper transit occurs on the opposite side of the zenith from the pole. In that case h must still represent tlie distance of the star from the point of the horizon below the pole, and will exceed 90°. Thus, among the Greenwich observations we find 1837 Juno 14, Capella h^ = 7° 18' 7".94 h = 95 39 7 .91 q> =bl 28 37 .93 163. Meridian zenith distances of the snn observed near the summer and winter solstices. — ^When the place of observation is near the equator, the lower culminations of stars can no longer be ob- served, and, consequently, the method of the preceding article cannot be used. The latitude found from stars observ^ed at their upper culminations only is dependent upon the tabular declina- tion, and is, therefore, subject to the error of this declination. If, therefore, an obsen^atory is established on or near the equator, and its latitude is to be fixed independently of observations made at other places, the meridian zenith distances of stars cannot be employed. The only independent method is then by meridian observ'ations of the sun near the solstices. Let us at first suppose that the obser\'ations can be obtained exactly at the solstice, and the obliquity (e) of the ecliptic is constant. The declination of the sun at the summer solstice is = -i- £, and at the winter solstice it is --^ — e ; hence, from the meridian zenith distances ^ and ^' observ^ed at these times, we should have y = :' — € the moan of which is sp = J (: + :') a result dependent only upon the data furnished by the obsen'a- tions. Xow, the sun will not, in general, pass the meridian of the observer at the instant of the solstice, or when the declination is at its maximum value e; nor is the obliquity of the ecliptic con- stant. But the changes of the declination near the solstices are very small, and hence are verj- accurately obtained from the ALTITUDE AT A GIVEN TIME. 229 solar tables (or from the Ephemeris which is based on these tables), notwithstanding small errors in the absolute value of the obliquity. The small change in the obliquity between two solstices is also verj' accurately known. If then Ae is the un- known correction of the tabular obliquity, and the tabular values at the two solstices are e and e', the true values are e + Ae and e' + AS ; and if the tabular declinations at two observations near the solstices are e — x and — (e' — x'), the true declinations will be 5 = e + Ae — X and 5' = — (e' + Ae — x'), and by the formula f> = ^ + 5 we shall have for the two observations ^ = C -{- S -\- ^€ X 9» = C' — e' — Ae + a:' the mean of which is ^ = i (c + o + J (^ - O - K^ - ^) a result which depends upon the small changes e — e' and x — x% both of which are accurately known. It is plain that, instead of computing these changes directly, it suffices to deduce the latitude from a number of observations near each solstice by employing the apparent declinations of the solar tables or the Ephemeris ; then, if <p' is the mean value of the latitude found from all the observations at the northern solstice, and f" the mean from all at the southern solstice, the true latitude will be Every observation should be the mean of the observed zenith distances of both the upper and the lower limb of the sun, in order to be independent of the tabular semidiameter and to eliminate errors of observation as far as possible. SECOND METHOD. — BY A SINGLE ALTITUDE AT A GIVEN TIME. 164. At the instant when the altitude is observed, the time is noted by the clock. The clock correction being known, we find the true local time, and hence the star's hour angle, by the fonnula t = e —a in which © is the sidereal time and a the star's right ascension. 230 LATITUDE. K the Bun is observed, i is simply the apparent solar time. We have, then, by the first equation of (14), sin f sin ^ -f cos f cos S cos t = sin A in which f is the only unknown quantity. To determine it, assume d and J) to satisfy the conditions d sin B = sin S d cos D = cos d cos t then the above equation becomes d cos (f — ^) = sin h which determines <p — D^ and hence also tp. For practical con- venience, however, put then, by eliminating d, the solution may be put under the follow- ing form : tan D = tan d sec t cos 7' =r sin A sin 2> cosoc d ) (281) <p = D dtr The first of these equations fully determines D, which will be taken numerically less than 90°, positive or negative according to the sign of its tangent. As t should always be less than 90°, or Q^y D will have the same sign as d. The second equation is indeterminate as to the sign of j^ since the cosine of + ;' and — y are the same. Hence we obtain by the third equation two values of the latitude. Only one of these values, however, is admissible when the other is numerically greater than 90°, which is the maximum limit of latitudes. When both values are within the limits + 90° and — 90°, the true solution is to be distinguished as that which agrees best with the approximate latitude, which is always suffi- ciently well known for this purpose, except in some peculiar eases at sea. Example 1. — 1856 March 27, in the assumed latitude 23° S. and longitude 43° 14' W., the double altitude of the 8un*8 lower ALTITUDE AT A GIVEN TIME. 231 limb observed with the sextant and artificial horizon was 114*^ 40' 30'' at 4* 21"* 15* by a Greenwich Chronometer, which wa^ fast 2r 30*. Index Correction of Sextant = — 1' 12", Barom. 29.72 inches, Att Therm. 61^ F., Ext. Therm. 61° F. Required the true latitude. Sextant reading = 114® 40' 30" Chronometer 4* 21"» 15- Index eorr. = — 1 12 Correction — 2 30 114 39 18 Gr. date, March 27, 4 18 45 App. alt. Q — 57 19 39 Longitude — 2 52 56 Semidiameter — + 16 3 Local mean t. — 1 25 49 Eef. and par. = — 31 Eq. of time — 5 19 A— 57 35 11 App. time, t = 1 20 30 a — +2 51 30 — 20° 7' 30" log sec t 0.027360 log tan d 8.698351 log tan D 8.725711 log cosec <J 1.302190 log sin D 8.725098 • D=+ 3® 2' 38" log sin A 9.926445 r = 25 58 49 log cos r 9.953733 D — 7- = SP = — 22 56 11 Example 2. — 1856 Aug. 22; suppose the true altitude of Fomalhmit is found to be 29° 10' 0" when the local sidereal time is 21* 49-* 44- ; what is the latitude ? We have a = 22* 49- 44-, whence f= — l*0-0';a= — 30^22' 47".5; i)= — 31° 15' 13", /- = db 60° 0' 6", 9) = + 28° 44' 53". The nega- tive value of Y hero gives <p = — 91° 15' 19"; which is inadmissible. 165. The observation of equal altitudes east and west of the meridian may be used not only for determining the time (Art. 139), but also the latitude. For the half elapsed sidereal time between two such altitudes of a fixed star is at once the hour angle required in the method of the preceding article. When the sun is used in this way, the half difterence between the apparent times of the observations is the hour angle, and the declination must be taken for noon, or more strictly for the mean of the times of observation. By thus employing the mean of the A.M. and P.M. hour angles and the mean of the corresponding declinations, we obtain sensibly the same result 232 LATITUDE. as by computing each observation separately with its proper hour angle and declination and then taking the mean of the two resulting latitudes; and an error in the clock correction does not aftect the final result. The clock rate, however, must be known, as it affects the elapsed interval. See also Art 182. 166. Effect of errors in the data upon the latitude computed from an observed altitude. — From the first of the equations (51) we find , dZ sinfl'cos^ ,^ , cos or ,^ dc> = ? — - — at A 5- dd cos A cos^l cos -A or, since h = 90° — ^^ dh = — rf^, and sin q cos d = cos f sin -4, d<p z= — sec A,dh — cos ^ tan A,dt -{■ cos q sec A . dd whence it appears that errors of altitude and time will have the least cftect when -4 = or 180°, that is, when the observation is in the meridian, and the greatest eftcct when the observation is on the prime vertical. K the same star is observed on both sides of the meridian and at equal distances from it, the coeffi- cient oi dt will have opposite signs at the two ob8er\'ation8, and hence a small error in the time will be wholly eliminated by taking the mean of the values of the latitude found from two such observations. It is advisable, therefore, in taking a series of observations, to distribute them symmetrically with respect to the meridian. Wlien they are all taken verj- near to the meri- dian, a special method of reduction is used, which will be treated of below as our Tlnrd Method of finding the latitude. The sign of sec A is difterent for stars north and south of the zenith: hence errors of altitude will be at least partially eliminated bv takinuj the mean of the results found from stars near the meridian, both north and south of the zenith. A constant error of the instrument may thus be \rholhj eliminated. As for the effect of the error do^ its coeflicient is zero onlv when q -- 90° and sec A is not infinite. Tliis occurs when a circumpolar star is obscned at its elongation, where we liave. Art. 18, , cos & QQQ yl — _: |. [sin {r, -f- y-) sin (<J — y-)] which shows that sec A diminishes as d increases. In order, therefore, to re<luce the cftect of an error in the declination REDUCTIOX TO THE MERIDIAN. 283 at the same time with that of errors of altitude and time, we should select a star as near the pole as possible, and observe it at or near its greatest elongation, on either side of the meridian. The proximity of the star to the pole enables us to facilitate the reduction of a series of observations, and we shall therefore treat specially of this case as our Fourth Method below. 167. When several altitudes not very far from the meridian are observed in succession, if we \vish to use their mean as a single altitude, the correction for second differences (Art. 151) must be applied. It is, however, preferable to incur the labor of a sepa- rate reduction of each altitude, as we shall then be able to com- pare the several results, and to discuss the probable errors of the observations and of the final mean. Wlien the observations are very near to the meridian, this separate reduction is readily effected, with but little additional labor, by the following method : THIRD METHOD. — BY REDUCTION TO THE MERIDIAN WHEN THE TIME IS GIVEN. 168. To reduce an altitude^ observed at a given time^ to the meridian. — This is done in various ways. (A.) If in the formula, employed in Art. 164, sin <p sin d -\- cos <p cos 5 cos i = sin A we substitute cos ^ = 1 — 2 sin' i t it becomes sin ^ sin ^ -f cos <p cos d — 2 cos f cos d sin' J ^ = sin A But sin ^ sin 5 -{- cos <p cos d = cos (^ — S) or cos (d — ip) Hence, if we put Cj = sp — 5, or d = ^ — s^ the above equation may be written cos C, = sin A + cos q> cos d (2 sin' i f) (282) If the star does not change its declination, ^j is the zenith distance of the star at its meridian passage ; and, being found by 23-4 LATITUDE. this equation, we then have the latitude as from a meridian observation by the formula SP = ^ + Ci, or ^ = a — Ci according as the zenith is north or south of the star. Wlien the star changes its declination, this method still holds if we take d for (he time of observaiion, as is evident from our formulae, in which S is the declination at the instant when the true altitude is h. To compute the second member, a previous knowledge of the latitude is necessaiy. As the term cos f cos 8 (2 sin* J t) de- creases with /, if the observations are not too far from the meridian, the error produced by using an approximate value of y will be relatively small, so that the latitude found will be a closer approximation than the assumed one ; and if the computa- tion be repeated wdth the new value, a still closer approximation may be made, and so on until the exact value is found. This method is only convenient where the computer is pro- vided with a table of natural sines and cosines, as well as a table of log. versed sines, or the logarithmic values of 2 sin' J L Example. — Same as Example 1, Art. 164. h = 57® 35' 11", J = + 2° 51' 30", i = 1* 20"' 30*. Approximate value of y = -- 23**. log (2 sin» J t) 8.785726 log cos <p 9.964026 nat. cos h 0.844201 log cos d 9.999459 nat. no. 0.050182 log 8.749211 nat. cos :, 0.000:^^3 :, = — 25° 47' 54" (zenith south of sun.) d --^ ^ 2 51 30 ^ = — 22 "50^4 diflering but 13" from the true value, although the assumed latitude was in eiTor nearly 4'. Kepeating the computation with — 22° 56' 24" as the approximate latitude, we find f = — 22° 56' 11", exaetlv as in Art. 164. 169. (B.) AVe may also compute directly the reduction of the objiened altitude to the meridian altitude. Putting A, z^ meridian altitude = 90° — C, it u CIRCUMMERIDIAN ALTITUDES. 2So the formula (282) gives sin h^ — sin A = 2 cos f cos <J sin" } t But we have sin Aj — sin A = 2 cos J (h^ + h) sin i (h^ — h) and hence sin } Ch. — A) = (283) ^ * "^ cos J (^1 + A) which gives the diflTerence Aj — A, or the correction of A to reduce it to Ai ; but it requires in the second member an approximate value both of tp and of A^, the latter being obtained from the assumed value of if by the equation A^ = 90° — (^ — S)\ or, if the zenith is south of the star, by the equation A^ = 90° — (5 — ^). Example. — Same as the above. a = 2^ 61' 30" logsin«i< 8.484696 Approx. f = — 23 00 00 log cos tp 9.964026 Cj = 26 61 30 log cos a 9.999459 \= 64 8 30 log sec i (A, + A) 0.312573 a J (^^ + A) = 60 51 50 log sin i (A, — A) 8.760754 Aj — A = 6 36 33 A = 57 35 11 a = 2° 51' 30" \= 64 1144 C^ = ~ 25 48 16 f = — 22 66 46 This method does not approximate so rapidly as the preceding, but the objection is of little weight when the observations are very near the meridian. On the other hand, it has the great advantage of not requiring the use of the table of natural sines. 170. (C.) CircumTneridian altitudes. — ^When a number of altitudes are observed very near the meridian,* they are called circum- meridian altitudes. Each altitude reduced to the meridian gives nearly as accurate a result as if the observation were taken on the meridian. An approximate method of reducing such observations with the greatest ease is found by regarding the small arc \^{h^ — A) as sensibly equal to its sine ; that is, by putting sin } (A, — A) = i (A, — A) sin 1" * How near to the meridian wiU be determined in Art. 175. 236 LATITUDE. and taking h^ for } {h^ + A), from which it difters very little, 80 that (283) may be put under the form , . co8c>co8^ 2 8in*}f /-ftojN ^1 — ^ = 1 T-^rjT (284) cos Aj sm 1" The value in seconds of 2 sin* i t VI = sin 1" is given in Table V. with the argument t. If h\ h'\ A'", &c. are the observed altitudes (corrected for refraction, etc.); <',<",/"', &c., the hour angles deduced from the observed clock times; m', m", m'", &c., the values of m from the table ; and we put the constant factor (285) . cos <p cos d cos ip cos d cos A, sin Cx we have A, = A' + Am' A, = A" + Aw!' \ = A'" + Am'" &c. and the mean of all these equations gives h' J- h" + A'" + etc. , . m' + m" + m'" + &c. A = 1- A n n in which n is the number of observations ; or h, = h^ + Am, (286) in which A^, denotes the mean of the observed altitudes corrected for refraction, &c., and m^ the mean of the values of m. When Aj has been thus found, the latitude is deduced as from any meridian altitude, only ob8er\'ing that for the sun the de- clination to be used is that which corresponds to the mean of the times of observation, as has already been remarked in Art. 168. Example. — At the U. 8. Naval Academy, 1849 June 22, cir- cummeridian altitudes of ^9 Ursac 3Iinoris were observed with a Troughton sextant from an artificial horizon, as in the following table. The times were noted by a sidereal chronometer which CIRCUMMERIDIAN ALTITUDES. 237 was fast 1* 45'.7. The index correction of the sextant was — 14' 58", Barometer, 30.81 inches, Att. Therm. 65° F., Ext. Therm. 64° F. The right ascension of the star was 14* 51* 14*.0 Chronometer fast +1 45 .7 Chronometer time of star's transit 14 52 59 .7 The hour angles in the column t are found by taking the differ- ence between each observed chronometer time and this chro- nometer time of transit. 2 Alt. :i|c Chronom. t fit 108O 89* 40" 14* 45^ 47». 7* 12».7 102.1 89 50 47 1. 5 58.7 70.2 40 40 48 54.5 4 5.2 82.8 41 51 29.5 1 80.2 4.4 41 54 86.5 1 86.8 5.1 40 80 56 22. 8 22.8 22.8 40 20 57 48. 4 48.8 48.8 40 58 47.5 5 47.8 66.0 40 15 17.5 7 17.8 104.5 89 20 2 10. 9 10.8 165.1 Mean 108 40 14 jii0 = = 61.68 Ind. corr. — 14 58 108 25 16 Assumed ^ — 88« 59*0" 54 12 88 6 — 74 46 86 .9 Refr. — 42.0 Approx. Ci = 85 47 86 .9 log COB ^ 9.8906 Am^ + 21 .5 log 008 6 9.4198 Ai- 54 12 17 .6 log cosecCi 0.2829 Ci-- - 86 47 42 .6 log A 9.5428 6 — 74 46 86 .9 log m^ 1.7898 ^ - 88 58 54 .4 log Am^ 1.8826 Remark 1. — The reduction h^ — h increases as the denominator of A decreases, that is, as the meridian zenith distance decreases. The preceding method, therefore, as it supposes the reduction to be small, should not be employed when the star passes very near the zenith, unless at the same time the observations are restricted to very small hour angles. It can be shown, however, from the more complete formulsB to be given presently, that so long as the zenith distance is not less than 10°, the reduction computed by this method may amount to 4' 30" without being in error more than 1" ; and this degree of accuracy suffices for even the best observations made with the sextant. 238 LATITUDE. Remark 2.— K in (284) we put sin J < = V sin V. t {t being in seconds of time), we have , - cos f cos ^ 225 . - „ ., ., .^^^ Ai — A = sm 1". ^' = at* (287) cos h^ 2 in which a denotes the product of all the constant factors. It follows from this formula that near the meridian the altitude varies as the square of the hour angle, and not simply in proportion to the time. Hence it is that near the meridian we cannot reduce a number of altitudes by taking their mean to correspond to the mean of the times, as is done (in most cases without sensible error) when the observations are remote from the meridian. The method of reduction above exemplified amounts to sepa- rately reducing each altitude and then taking the mean of all the results. 171. (D.) Circummeridian altitudes more accurately reduced. — The small correction which the preceding method requires will be obtained by developing into series the rigorous equation (282). This equation, when we put ^ = 90° — h = true zenith distance deduced from the observation, may be put under the form cos C = cos Ci — 2 cos ^ cos d sin* J t which developed in series* gives, neglecting sixth and higher powers of sin J /, * If we put y = 2 cos ^ cos 6 sin' } t, the equation to be dcTeloped is cos c = cos Ci — y (a) in which ^^ is constant and ^ may be regarded as a fiinotion of y ; so thai bj Mac- LAUBiN^s Theorem in which (/), | — j. &c. denote the Talues of /y and its differential coefficient* when y ^=r 0. The equation (a) gires, by differentiation, sin ^ — =1 — = </y rfy sin ^ rf*C COS C <^ cot c rfy' sin' ^ dy sin» ^ &0. COS sin CIRCUMMEBIDIAN ALTITUDES. 239 ip COS ^ 2 sin* \ t I cos y cos d \« 2 cot Ci sin* j t . g ;hr^^ sinl" I sinCi / sin 1" By this formula, first given by DelambRb, the reduction to the meridian consists of two terms, the first of which is the same as that employed in the preceding method, and the second is the small correction which that method requires. These two terms will be designated as the " Ist Eeduction" and " 2d Reduction." Putting we have 2 sin« i t m — sin 1" 2 sin' } t n — sin 1" - cos cos d A — B — A* cot Ci sin Ci Ci c — Am + Bn (289) If a number of observations are taken, we have a number of equations of this form, the mean of which will be Ci = Co — Am^ + Bn^ in which ^j, is the arithmetical mean of the observed zenith dis- tances, niQ and n^ the arithmetical means of the values of 7n and 71 corresponding to the values of t. The values of n are also given in Table V. Having found (^j, we have the latitude, as before, by the formula ^ = ^ + C, in which we must give (^^ the negative sign when the zenith is south of the star, and it must be remembered that for the sun (or any object whose proper motion is sensible) 8 must be the mean of the declinations belonging to the several observations. But when y = we have, by (a), ^ = Ci> bo that (6) becomes y y' cot ^, «• BeBtoring the Talue of y, this giYes the deyelopment used in the text, obseiring that as Z and ^^ are supposed to be in seconds of arc, the terms of the series are divided by sin 1'' to reduce them to the same unit. 240 LATITUDE. or, which is the same, the declination corresponding to the mean of the times of observation.* Finally, if the star is near the meridian below the pole, the hour angles should be reckoned from the instant of the lower transit. Eecurring to the formula cos C = sin ^ sin d -\- cos f cos d cos t in which t is the hour angle reckoned from the upper transit, we observe that if this angle is reckoned from the lower transit we must put 180° — i instead of /, or — cos i for + cos U and theu we have cos C = sin ^ sin d — cos f cos d cost and, substituting as before, cos i = 1 — 2 sin" } t this gives cos C = — cos (f + ^) + 2 cos f cos d sin" i t or, since for lower culminations we have f^ = 180*^ — (s^ + ^ and cos ^^ = — cos (^ + 5), cos C = cos C, + 2 cos f cos d sin* J t which developed gives ^ , cos ^ cos d 2 sin* it / cos f cos d^ 2 cot Ti sin* } t * * sin Ci sin 1" \ sin Ci / sin 1" or Ci = C + ilm + J?n (sub polo) (290) which is computed by the same table, but both first and second reductions here have the same sign. If a 8tar is obsen-ed with a sidereal chronometer the daily rate of which is so small as to be insensible during the time of * To Bhow that the mean declination is to be used, we maj obserre that for each observation we have put Ci = ^ — (5, and that if d\ d'\ &c., are the seTenJ deelin*- tions, the seTeral equations of the form (289) will give ^ = <J' 4- ;' — ^m' -f A* cot ^, n' ^ = <J" + ^' — Am'f -f A^ cot ^j n" &c., the mean of which, if <J = mean of S\ 6'\ &c., will be ^ = <J + (• — -4wo H- -i* cot Ci fi, == <J -f. fj CIRCUMMERIDIAN ALTITUDES. 241 the observations, the hour angles i are found by merely taking the difference between each noted time and the chronometer time of the star's transit, as in the example of Article 170. But if we wish to take account of the rate of the chronometer, it can be done without separately correcting each hour angle, as fol lows: Let 8 The the rate of the chronometer in 24* (57 being positive for losing rate. Art 137) ; then, if < is the hour angle given directly by the chronometer, and i' the true hour angle, we have f:t = 24»: 24»— dT= 86400' : 86400- — dT whence f = t L S6400j Instead of sin J< we must use sin J<'; for which we shall have, with all requisite precision, sin } ^ = sin } ^ . — , or sin* if = sin' } ^ . j - j Hence, if we put we shall have . , cos cp cos ^ 2 sin' } t sm Ci sm 1" so that if we compute A by the formula cos f cos d A=:k' sin Ci we can take m = — ; — :rr- for the actual chronometer intervals, sm 1" ' and no further attention to the rate is required. The factor k can be given in a small table with the argument "rate," in connection with the table for m, as in our Table V. K a star is observed with a mean time chronometer, the inter- vals are not only to be corrected for rate, but also to be reduced Vol. L— 1« 242 LATITUDE. from mean to sidereal intervals by multiplying them by ;£ = 1.00273791 (Art. 49) ; so that for sin* i t we must substitute k sin* (J./i^), or, with sufficient precision, kff sin* } L K the sun is observed with a mean time chronometer, the in- tervals are both to be corrected for rate and reduced from mean solar to apparent solar intervals. The mean interval differs from the apparent only by the change in the equation of time during the interval, and this change may be combined with the rate of the chronometer. Denoting by 8E the increase of the equation of time in 24* (remembering that -E is to be regarded as positive when it is additive to apparent time), and by ^Tthe rate of the chronometer on mean time, we may regard dT — dE as the rate of the chronometer on apparent time. Instead of the factor k we shall then have a factor A;', which is to be found by the formula L "~ 86400 which may be taken from the table for k by taking dT — dE as the argument. Finally, if the sun is observed with a sidereal chronometer, we must multiply sin* J t not only by A' but by -\, Denoting // by i and — , by t', these rules may be collected, for the convenience of reference, as follows: Star by sidereal chron., ^ = A* . -, fiin Cj Star b\' mean time cbron.,^ = Ai — ' .^^ [log t= 0.002375] sin C, (291) r, 1 .1 J f . COS <r cos r5 Sun by mean time chron., A =: k ; sin Ci Sun by sidereal chron., A=^h! i' ^^'1?L5^1_ [log a = 9.997625] sin «»| for which log k will be taken from Table V. with the argument rate of the chronometer = dT\ and log k' from the same table CIRCUMMERIDIAN ALTITUDES. 243 with the argnment dT — dE=^ daily rate of the chronometer diminished by the daily increase of the equation of time. Example. — 1856 March 15, at a place assumed to be in lati- tude 87° 49' N. and longitude 122° 24' W., suppose the fol- lowing zenith distances of the sun's lower limb to have been observed with an Ertel universal instrument,* Barom. 29.85 inches, Att Therm. 65° F., Ext. Therm. 63° F. The chrono- meter, regulated to the local mean time, was, at noon, slow 11- 20*.8, with a daily losing rate of 6'.6. m n Obs'd xeD. dist. Chronometer. t 40° 8'40".7 23»37-85*. — 19«58'.8 788".8 1".49 40 2 16.5 42 3. —15 80.8 472.4 0.54 39 57 28 .8 46 29.5 —11 4.8 240 .6 .14 89 54 17 .2 50 46.5 — 6 47.8 90 .5 .02 89 52 88 . 55 16. — 2 17.8 10 .4 .00 89 52 84 .5 87 .5 + 8 8 .7 18 .4 .00 89 54 28 .6 5 18 . 7 89 .2 115 .0 .08 89 58 9 .8 9 49.5 12 15.7 295.1 0.21 40 8 .8 14 8 . 16 34 .2 588 .9 .70 40 9 86 . 18 81 . 20 57 .2 861 .4 1 .80 Means 89 59 18 .5 U= +0 29.1 7?io=842 .60no=0 .49 The equation of time at the local noon being + 8* 54*.6, we have Mean time of app. noon = 0* 8* 54*.6 Chronometer slow = 11 20.8 Chr. time of app. noon = 28 57 88 .8 The difference between this and the observed chronometer times gives the hour angles t as above. The mean of the hour angles being + 29*.l, the declination is to be taken for the local apparent time 0* 0* 29^.1, or for the Greenwich mean time March 15, 8* 18"* 59^.7; whence ^ = — 1<> 48' 8".8 (Approximate) ^ = -f- 87 49 . " Ci= 89 87 8 .8 The increase of the equation of time in 24* is dE = — 17'.4, * See Vol. IL, Altitude and Azimuth Instrument^ for the method of obserTing the senith dieianoes. 244 LATITUDE. and, the chronometer rate being 8T = + 6\6, we have dT — 8E =^ + 24*. 0, with which as the argument "rate" in Table V. we find log A' = 0.00024. The computation of the latitude is now carried out as follows: log cos f 9.89761 Mean observed zen. dist. Q = 39® 59^ 18''.5 log cos d 9.99979 r—p= + 41 .8 log cosec :, 0.19540 log A* 0.1861 S= — 16 6 .5 log ^ 0.00024 log cot C, 0.0821 ^iWp = — 7 4 .4 log^ 0.09304 logJB 0.2682 Bno= + .9 log wio 2.53479 log no 9.6902 C, = 39 36 60 .3 logilmo 2.62783 log J?no 9.9584 d= — 1 48 8 .8 f = 37 48 41 .5 The assumed value of y being in error, the value of A is not quite correct; but a repetition of the computation with the value of <p just found does not in this case change the result so much as O'M. 172. (E.) Gauss's method of reducing circummeridian altitudes of the sun. — The preceding method of reduction is both brief and accurate, and the latitude found is the mean of all the values that would be found by reducing each ob8er^'ation separately. This separate reduction, however, is often preferred, notwith- standing the increased labor, as it enables us to compare the observ'ations with each other, and to discuss the probable error of the final result; and it is also a check against any gross error. But, if we separately reduce the observations by the preceding method, we must not only inteqiolate the refraction for each altitude, but also the declination for each hour angle. Gavss proposed a method by which the latter of these inteqmlations is avoided. He showed that if we reckon the hour angles, not from apparent noon, but from the instant when the sun reaches its warinmm altitude^ we can compute the reduction by the method above iriven, and use the meridian declination for all the obserx'a- tions. This method is, indeed, not quite so exact as the preced- ing; but I shall show how it nniy be rendered (piite perfect in practice by the introduction of a small correction. In the rigorous formula cos C = sin <f sin H + cos f cos d cos t CIRCUMMERIDIAN ALTITUDES. 245 d is the declination corresponding to the hour angle t. If then aJ = the hourly increase of the declination, ^positive when the sun is moving northward^ d^ = the declination at noon, and if ^ is expressed in seconds of time, we have t . Ad * ^ 8600 ' ^ where, since ^d never exceeds 60", x will not exceed 80" so long as < < 80*. Hence we may suhstitute, with great accuracy, sin d = sin d^ -|- cos d^ sin x cos J = cos dj — sin d^ sin a: and the ahove formula becomes cos C = sin 5? sin d^ + cos f cos ^^ cos f + sin (^ — d^) sin x 4- 2 COB f sin dj sin' } ^ sin or The last term is extremely small, rarely affecting the value of ^ by as much as 0".l ; and since x is proportional to the hour angle, and therefore has opposite signs for observations on differ- ent sides of the meridian, the effect of this term will nearly or quite disappear from the mean of a series of observations pro- perly distributed before and after the meridian passage. Now, we have fAdsinl" i;:. . i« Ad sm x = — --— — = 15 ^ sm 1" . Let then, taking 8600 54000 . « Ad Bin(cp — O sm ^ = — ^ ^ 54000 cos f> cos d^ 15 t sm 1" = sin f + J sin» t we have smx =(sm t + i sin' t) sin i^ • — sin (^ — dj and the formula for cos ^ becomes, by omitting the last term, cos C = sin ^ sin d^ -\- cos f cos dj(co8 1 + sin t sin t>) -(- } cos ^ cos d^ sin'^ sin t9 246 LATITUDE. The last temi involving sin' t multiplied by the small quantity sin I? is even less than the term above rejected. Like that, also, it has opposite signs for observations on different sides of the meridian, and will not aftect the mean result of a properly arranged series of observations. Rejecting it, therefore, our for- mula becomes cos C = sin f sin d^ -f cos ^ cos d^ cos (t — d) -f 2 cos ^ cos d^ sin* J d^ The last term here must also be rejected if we wish to obtain the method as proposed by Gauss ; but, as it is always a positive term and affects all the observations alike, I shall retain it, espe- cially as it can be taken into account in an extremely simple manner. The maximum value of cos ^, which corresponds to the maximum altitude, is given immediately by the above formula by putting t = t?. Hence t? is the hour angle of the maximum altitude. Putting we have Let cos C = cos (f — d^) — 2 cos f> cos d^ sin* } f + 2 cos ^ cos d^ sin* J * , cos f cos d 2 sin* 1 ^ 0=0^ -j *-. sin (^ — d) sin 1" then our formula becomes cos C = cos (f — 5') — 2 cos f> cos d^ sin* J f This equation is of the same form as that from which (288) was obtained, and therefore when developed gives cos ^ cos 5, 2 sin* If / cos tp cos ^, \* 2 cot Cj sin* } f * sin Cj sin 1" \ sin Cj / sin 1" in which Ci = ^ — ^'- Putting then, as before, . cos cp cos d. _ ^. . - ,/*«.^x A = f— — 1 B = A* cot Ci (292) sm ^ 1 and taking m and ?i from Table V., or their logarithms from Table VI., with the argument t\ which is the hour angle reckoned CIKCUMMERIDIAN ALTITUDES. 247 from the instant the sun reaches its maximum altitude, we have Ci = C — Am + Bn (293) Since (^^ differs from the latitude by the constant quantity 5', its value found from each observation should be the same. Taking its mean value, we have f = C, + ^' The angle t>, being very small, may be found with the utmost precision by the formula ^ = — = [9.40694] — (294) 810000 sin r A *• ^ A ^ which gives ^ in seconds of the chronometer when A has been computed by the formula (292). The most simple method of finding the corrected hour angles V will be to add t? to the chronometer time of apparent noon, and then take the difference between this corrected time and each observed time. If we put d^ = di + y, we have y = A.l^ (295) Sin 1" which requires only one new logarithm to be^taken, namely, the value of log m from Table VI. with the argument &. We then have, finally, 9 = ^^+^i + y (296) Example. — The same as that of the preceding article. We have there employed the assumed latitude 37® 49' ; biit, since even a rough computation of two or three observations will give a nearer value, let us suppose we have found as a first approxima- tion y = 37° 48' 45". With this and the meridian declination *i = — 1° 48' 9".2, and log A' = 0.00024 as before, we now find, by (292), log A = 0.09810 log B = 0.2683 We have also there found the chronometer time of apparent 248 LATITUDE. noon = 23* 57* 33\8. We now take from the Ephemeris Ai = + 59".22, and hence, by (294), log Ad 1.7725 ar. CO. log A 9.9069 const, log 9.4059 ^ = + 12'.2 log * 1.0853 Hence the chronometer time of the maximum altitude is 23* 57" 33'.8 + 12'.2 = 23* 57" 46', which gives the hour angles /' at3 below : a logm log Am logn logSn — 20- 11*. 2.90274 2.99584 0.1900 0.4583 15 43. 2.68558 2.77868 9.7557 0.0240 11 16.5 2.39718 2.49028 9.1776 9.4459 6 59.5 1.98216 2.07526 8.8487 8.6170 — 2 30. 1.08891 1.18201 + 2 51.5 1.20525 1.29835 7 27. 2.03730 2.13040 8.4553 8.7286 12 3.5 2.45551 2.54861 9-2955 9.5638 16 22. 2.72077 2.81387 9.8260 0.0943 20 45. 2.92677 3.01987 0.2381 0.5064 The refraction may be computed from the tables first for a mean zenith distance, and then with its variation in one minute (which will be found with sufficient accuracv from the table of mean refraction) its vahie for each zenith distance is readily found. The parallax, which is here sensibly the same {= 6".54) for all the observations, is subtracted from the refraction, and the results are given in the column r — p of the following computation. The numbers in the 3d and 4th columns are found from their logarithms above ; and the last column contains the sevenU values of the minimum zenith distance of the sun's lower limb, formed by adding together the numbers of the preceding columns. To the mean of these we then apply the sun's semidiameter, the meridian declination, and the correction y, which are all constant for the whole series of observations. CIRCUMMERIDIAN ALTITUDES. 249 Obs'd ; T—p An Bn < •1 40<» 8'40".7 + 42".l — 16' 30".5 + 2".9 39* > 52' 55".2 40 2 16 .5 41 .0 10 .7 1 .1 58 .8 89 57 28 .3 41 .8 5 9 .2 .8 61 .2 89 54 17 .2 41 .7 1 58 .9 .0 60 .0 89 52 33 . 41 .6 15 .2 .0 59 .4 89 52 34 .5 41 .6 19 .9 .0 56 .2 89 54 28 .6 41 .7 2 15 .0 .1 55 .4 39 58 9 .8 41 .8 5 58 .7 .4 58 .3 40 3 .3 41 .9 10 51 .4 1 .2 52 .0 40 9 36 . 42 .1 17 26 .8 8 .2 54 .5 (Lower limb) Mean C, — 39 52 57 .10 , 2sin»i* '"« sinl" 8.9090 Semidiameter =: 16 6 .49 *.- — 1 48 9 .20 log^ 0.0931 y — + .10 logy 9.0021 9 — 37 48 41 .51 This result agrees precisely with that found before. If we suppose all the obsei-vations to be of the same weight, we can now deter- mine the probable error of observation. Denoting the difference between each value of {^^ and the mean of all by r, and the sum of the squares of v by [fr], according to the notation used in the method of least squares, we have V vv — 1".9 8.61 + 1.7 2.89 + 4.1 16.81 + 2 .9 8.41 + 2 .3 5.29 — .9 .81 — 1 .7 2.89 + 1.2 1.44 — 5 .1 26.01 -2 .6 6.76 Mean error of a single obscrva- tion = Ji?^l = 2".89 A/n — 1 Mean error of the final value of 2^ ^ ^,^^ l/lO n = 10, [vv'] = 74.92 Probable error of a single obs. = 2".89 X 0.6745 = 1".95 « « of sp =0 .91 X 0.6745 = .61 It must not be forgotten that the probable error 1".95 here represents the probable error of observation only : a constant error of the instrument, affecting all the obser\^ations, will form no part of this error; and the same is true of an error in the refraction. 250 LATITUDE. 173. For the most refined determinations of the latitude, standard stars are to be preferred to the sun. Their deelinatious are somewhat more precisely known ; the instrument is in night observations less liable to the errors produced by changes of temperature during the observations ; constant instrumental errors and errors of refraction may bo eliminated to a great extent by combining north and south stars ; or errors of declina- tion may be avoided by employing only circumpolar stars at or near their upper and lower culminations. In general, errors of circunmieridian altitudes are eliminated under tlie same condi- tions as those of meridian observations, since the former are reduced to the meridian with the greatest precision. See the next following article. For a great number of nice determinations of the latitude by circummeridian altitudes of stars north and soutli of the zenith and of circumpolar stars, see Puissant, Nouvclle Dcsmption Gio- mitriquc de la France. 174. Effect of errors of zenith distance^ declination^ and timCj upon the latitude found by circunmieridian altitudes. — Difterentiating (289), regarding A as constant, and neglecting the variations of the last term, which is too small to be sensibly affected by small errors of /, we have, since dip —- rf^^i + ddy d^ = d: + dd- -4^"-^ (Ibdt) sm 1" The errors rf^ and dS aftect the resulting latitude by their whole amount. The coefficient of dt bus opposite signs for east and west hour angles; and therefore, if the observations are so taken as to consist of a number of pairs of equal zenith distances east and west of the meridian, a small constant error in tlie hour angles, arising from an imperfect clock correction, will be elimi- nated in the mean. This cimdition is in jjractice nearly satisfied when the same number of observations are taken on each side of the meridian, the intenals of time between the successive observations being made as nearly equal as practicable. An error in the assumed latitu<le which atl'ei^ts A is eliminated by repeating the computation with the latitude found by the first computation. An error in the declination which would affect A is not to be supposed. LIMITS OF CIRCUMMERIDIAN ALTITUDES. 251 175. To determine the Umiis within which the preceding methods of reducing circunimeridian altitudes are applicable. — First. lu the method of Art. 170 we employ only the " first reduction*' (= Am\ which is the first term of the more complete reduction expressed by (288). The error of neglecting the " second reduction** (= Bn) increases with the hour angle, and if this method is to be used it becomes necessary to determine the value of the hour angle at which this reduction would be sensible. We have -D .I. X ^ 2 sin* J* Bn = A* cot C, ^— ' sinl" whence if we put 5 for Bn and 2?' = l/TsmTManT we derive sin« it = — i/b (298) A Since Zi = f — 5, F and A are but functions of ^ and d ; and therefore by this formula we can compute the values of / for any assigned value of b, and for a series of values of ^ and 8. Table Vn.A gives the values of / in minutes computed by (298) when 5 = 1". That is, calling /j the tabular hour angle and t the hour angle for any assigned limit of error 6, we have sin* it^ = — sin' i t = sin' i t^ y^b A As the limits are not required with great precision, we may sub- stitute for the last equation the following : t = t//b If we take b = 0".l, we have yb = 0.56, or nearly J : hence the limiting hour angle at which the second reduction amounts to O'M is about } the angle given in Table VILA. Example. — How fur from the meridian may the observations in the example p. 237 be extended before the error of the method of reduction there employed amounts to 1"? With f= + 39°, d= + 75°, Table vri.A gives /j = 30^ Hence 252 LATITUDB, the method is in that example correct within 1" if the obserra- tions are taken within 30" of the meridian, and correct within O'M if they are taken within 15* of the meridian. Second. — ^In the more exact methods of reduction given in Arts. 171 and 172, we have neglected the last term of the development given in the note on page 239, which may be called a " third reduction." Denoting it by e, we have c = -- 1 — ■ ^ I J.' sm* i t 3 \ sin 1" / whence, if we put we deduce \l + 3cot«C, 8in« ht = — i/c (299) A Table Vil.B gives the values of ^ computed by this formula, for c = 1". Denoting the tabular value of t by /^ we have sin* } f, = — sin* it = sin* i t^ {/c A or, with sufficient accuracy in most cases, t = t^ V'C • / For c = 0".l we have i. e = 0.68, or nearly \ ; and hence the limiting hour angle at which the thirtl reduction (omitted in our most exact methods) would amount to O'M is about § the angle given in Table VII.B. Example. — How for from the meridian may the observations in the example p. 243 be extended before the error of the method of reduction there employed amounts to O'M ? With ip = 38^, a = — 2°, Table VII.B gives t^ -^ 39*, and § of this is / = 26": so that the method is in that example correct within 1" when the obsen-ations are taken within 31>" of the meridian ; and it is correct within 0".l when the obser\'ations arc taken \\'ithin 26"* of the meridian. The limiting hour angle for a given limit of error diminibhea BY THE POLE STAR. 258 rapidly with the zenith distance ; and hence in general very small zenith distances are to be avoided. But the observations may be extended somewhat beyond the limits of our tables, since the errors which affect only the extreme observations are reduced in taking the mean of a series. FOURTH METHOD. — BY THE POLE STAR. 176. The latitude may be deduced with accuracy from an alti- tude of the pole star observed at any time whatever, when this time is known. The computation may be performed by (281); but when a number of successive observations are to be reduced, the following methods are to be preferred. If we put p = the star's polar distance, we have, by (14), sin A = sin ^ cos p -j- cos ^ sin p cos t in which the hour angle t and the altitude h are derived from observation and <p is the required latitude. Now, p being small (at present less than 1° 30'), we may develop y in a series of ascending powers of py and then employ as many terms as we need to attain any given degree of precision. The altitude cannot differ from the latitude by more than 2); if, then, we put ^= h — X X will be a small correction of the same order of magnitude as p. We have* sin ^ = sin (h — ar) = sin h — x cos h — ^ x* sin A + J ^ cos h + &c. cos f> = cos (h — x) = cos h-^- XQinh — J x* cos h — J a;* sin A + &c. sinp =:p — J2^ + &c. cosp = 1 — Ip* + &c* which substituted in the above formula for sin h give sin A = sin A — x cosh -^ p cost cosh — ^(o:* — 2 xp cos t-\-p^) sin h-\-&c. and from this we obtain the following general expression of the correction : ♦ PL Trig. (408) and (406). 254 LATITUDE. a: = jp COS f — ^ (.r*— 2 xp COS f + ;>') tan A + 24 (^— 4 x»|> COS f + 6 x*/)*— 4 jy^cos t+p*) tan A — &c. (a) For a first approximation, we take x = pco&t (b) and, substituting this in the second term of (a), we find for a second approximation, neglecting the third powers of p and x, X = p cos t — J j>* sin* t tan A (c) Substituting this value in the second and third terms of (a), we find, as a third approximation, X = p cos t — Ip^ sin* t tan A + i />• cos t sin* t (d) This value, substituted in the second, third, and fourth terms of (a), gives, as a fourth approximation, X == p cos t — i j>*sin*< tan A + jp'cos t sin't — |/)*sin*f tan'A 4- ^1^ p4 (4 _ 9 sin. ^) gjn* < tan A (e) In these formulje, to obtain x in seconds when p is given in seconds, we must multiply the terms in /)*, pl^y and p* by sin 1", sin* 1", sin' 1", respectively. In order to determine the relative accuracv of these formulae, let us examine the several terms of the last, which embraces all the others. The value of t, which makes the last term of (c) a maximum, will be found by putting the difterential coefficient of (4 — 9 sin* /) sin* t equal to zero ; whence 4 sin ^ cos t (2 — 9 sin* = which is satisfied by ^ = 0, / = 90°, or sin* t = J, the last of which alone makes the second difterential coefticient negative. The maximum value of the term is, then, ^p* sin' l"tan A, which for p = 1° 30' = 5400" is 0".0018 tan A. This can amount to 0".01 only when A is nearly 80°, — that is, when the latitude is nearly 80°. This term, therefore, is wholly inappreciable in every practical case. BY THE POLE STAR. 255 The temi }2>*8in*l" sin^^tan' Ais a maximum for 8in< = l, in which case, for p = 5400", it is 0".0121 tan* h. This amounts to O'M when h = 64^, and to 1". when h = 77°. For the maximum of the term J jj^ sin^ 1" cos / sin* / we have, by putting the differential coefficient of cos t sin* t equal to zero, sin ^ (2 ~ 3 sin* t) = which gives sin* < = f , and consequently cos / = i/ J ; and hence the maximum value of this term is JjD^sin* 1"|/J = 0".475. Since the maximum values of this and the following terms do not occur for the same value of /, their aggregate will evidently never amount to 1" in any practical case. Hence, (o find the latitude by the pole star tmthin 1", we have the formula ^ = A — ;) cos f + i;)* sin 1" sin* t tan h (300) The hour angle t is to be deduced from the sidereal time of the observation and the star's right ascension a, by the formula ^== e — a To facilitate the computation of the formula (300), tables are given in every volume of the British Nautical Almanac and the Berlin Jahrbuch; but the formula is so simple that a direct computation consumes very little more time than the use of these tables, and it is certainly more accurate. Example. — (From the Nautical Almanac for 1861). — On March 6, 1681, in Longitude 37° W., at 7* 43"' 35* mean time, suppose the altitude of PolariSj when corrected for the error of the in- strument, refraction, and dip of the horizon, to be 46° 17' 28" : required the latitude. Mean time 7* 43- 35*. Sid. time mean noon, March 6, 22 56 47.9 Reduction for 7* 43« 35* 1 16.2 Reduction for Long. 2* 28* 24.3 Sidereal time e ^ — • 6 42 3.4 March 6, ;> — l'^ 25' 32".7 a = 1 7 39.0 t 5 34 24.4 83^36' 6" 256 LATITUDE. Hence, by formula (300), logp 3.71035 log|>« 7.4207 log cost 9.04704 logsin*^ 9.9946 log Ist term 2.75739 log tan h 0.0196 log i sin I'' 4.3845 h = 46° 17' 28" log 2d term 1.8194 Ist term = — 9 32 .0 2d « = + 1 6 .0 SP = 46 9 2 .0 By the Tables in the Almanac, 9> = 46** ^ 1" 177. If we take all the terms of (e) except the last, which we have shown to be insignificant, we have the formula f = A — p cos ^ + ll>* sin 1" sin' t tan h ^\f sin« 1" cos t sin« t + J;)* sin» 1" sin* t ten* A (301) which is exact within 0".01 for all latitudes less than 75®, and serves for the reduction of the most refined observations with first-class instruments. If we have taken a number of altitudes in succession, the separate reduction of each by this formula will be very laborious. To facilitate the operation, Petersen has computed verj^ con- venient tables, which are given in Schumacher's Hiilfstafdn^ edited by Warnstorff. These tables give the values of the following quantities : a=p^ cos t -\- \p^ sin* 1" cos t sin* t P =^iPo^ sin 1" sin* t X = Ip (;>* — ;>o*) ®*"* ^" ^^^ ' ®*^* ^ ;i =1 ^p* sin* 1" sin* t tan* h in which p, = r 30' = 5400". Then, putting Po log A = logp — 3.7323938 the formula (301) becomes ^ = h — (^Aa + X) + A*?ianh + /i BY TWO ALTITUDES. 257 Putting then we have ' ^ ^ f, = h'^x + y or, when the zenith distance ^ is observed^ X =z Aa, -\- X y = ^«/9cotC + A >(303) The first table gives a with the argument t ; the second, /? with the argument /; the third, X with the arguments p and t; and the fourth, /£ with the arguments y and y, ^ being used for h in BO small a term. To reduce a series of altitudes or zenith distances by these tables, we take for A or f the mean of the true altitudes or zenith distances ; for a and /? the means of the tabular numbers corresponding to the several hour angles, with which we find Aa and A^^ cot f . The mean values of the very small quanti- ties X and /£ are sensibly the same as the values corresponding to the mean of the hour angles ; so that X is taken out but once for the arguments polar distance and mean hour angle, and /jl is taken with the arguments f and the approximate value of y = A*^ cot ^. Illustrative examples are given in connection with the tables. FIFTH METHOD. — BY TWO ALTITUDES OP THE SAME STAR, OR DIP. FERENT STARS, AND THE ELAPSED TIME BETWEEN THE OBSERVA- TIONS. 178. Let S and S'y Fig. 25, be any two points of the celestial sphere, Z the zenith of the observer, Pthe pole. Suppose that the altitudes of stars seen at S and iS', either at the same time or diff^ercnt times, are observed, and that the observer has the means of determining the angle SPS' ; also that the right ascensions and declinations of the stars are known. From these data we can find both the latitude and the local time. A graphic solution (upon an artificial globe) is indeed quite simple, and it will throw light upon the analytic solution. With the known polar distances of the stars and the angle SPS'^ Vol. L— it 258 . LATITUDE. let the triangle SPS' be constructed. From S and S' as poles describe small circles whose radii (on the surface of the sphere) are the given zenith distances of /Sand S' : these small circles inter- sect in the zenith ^ of the observer, and, consequently, determine the distance PZ, or the co-latitude, and at the same time the hour angles ZPS and ZPS\ from either of which with the star's right ascension we deduce the local time. This graphic solution shows clearly that the problem has, in general, two solutions ; for the small circles described from S and S' as poles intersect in two points, and thus determine the zenith of another observer who at the same instants of time might have observed the same alti- tudes of the same stars. The analytic solution will, therefore, also give two values of the latitude; but in practice the ob- server always has an approximate knowledge of the latitude, which generally suffices to distinguish the true value. Let us consider first the most general case. (A.) T\co different stars observed at dfferent times. — ^Let A, h' = the true altitudes, corrected for refraction, &c., T, T' =: the clock times of observation, A T, A T' = the corresponding corrections of the clock, a, a' = the right ascensions, and d, d' = the declinations of the stars at the times of the observations respectively, tjf = the hour angles of the stars at the times of the observations respective!}', X = f—t = the difference of the hour angles, ^ = the latitude of the observer : then we have, if the clock is sidereal, a a f= r+ Ar— » X =(r— T) + (Ar'— aT)— (a'— a) (804) a formula which does not require a knowledge of the absolute values of A 7" and a 7"', but only of their difference; that is, of the rate of the clock in the interval between the two obser- vations. If the clock is regulated to mean time, the interval T' — T+ A y — A 7" is to be converted into a sidereal interval by adding the acceleration. Art. 49. We have next to obtain formulae for determining f and i or (' BY TWO ALTITUDES. 259 from the data A, A', i, ^', and X. I shall give two general solu- tions, the first of which is the one commonly known. Mrst SohUion. — ^Let the observed points S and S^ be joined by an arc of a great circle SS'. In the triangle PSS^ there are given the sides PS= 90^ — 8, PS' = 90° — 3', and the angle SPS' = Xj from which we find the third side SS' = -B, and the angle PS'S= P, by the formulse [a of Art. 10] cos B = sin d' sin d -{- cos d' cos d cos X sin B cos P == cos d' sin d — sin ^' cos ^ cos A sin J9 sin P e= cos S sin X or, adapted for logarithmic computation, m sin M= sin d m cos M= cos d cos A cos 5 = m cos (Jtf" — a') ) (305) sin 5 cos P = m sin (M — d^) sin 5 sin P = cos d sin A In the triangle ZSS' there are now known the three sides Z8 = 90*^ — A, jaS' = 90*^ — A', /Sfif' = P, and hence the angle ZS'S = Qy by the formula employed in Art. 22 : A/\ cos A' sin 5 / ^ ^ Now, putting ff = P-« there are known in the triangle PZS' the sides PS' = 90® — 5', ^^' = 90*^ — A', and the angle PS'Z = q, from which the side PZ= 90*^ — f , and the angle S'PZ= i\ are found by the formul» sin f> = sin d^ sin A' + cos ^' cos A' cos q cos f> cos ^ = cos d^ sin A' — sin d' cos A' cos q cos f sin f = cos A' sin g^ or, adapted for logarithmic computation, n sin iV = sin A' n cos JV= cos A' cos q sin sp = n cos (JVT— ^0 ) (307) cos 9> cos f = n sin (iV — d') cos f sin t' = cos A' sin q 260 LATITUDS. The fonnulsB (305) and (307) leave no doubt as to the quadrant in which the unknown quantities are to be taken. But we may take the radical in (306) with either the positive or the negative sign, and J §, therefore, either in the first or fourth quadrant. If we take J Q always in the first quadrant, the values of q will be and either value may be used in (307) ; whence two values of f and t'. That value of y, however, which agrees best with the known approximate latitude is to be taken as the true value. There is also another method of distinguishing which value of q will give the true solution ; for, if A^ and A are the azimuths of S' and Sj we have in the triangle ZSS' the angle SZS' = A' — A, and sm Q = sm (A' — A) ^ ^ ^ sin 5 in which cos h and sin B are always positive. Hence Q is posi- tive or negative according as -4' — A is less or greater than 180°. The observer will generally be able to decide this : the only eases of doubt will be those where A^ and A are very nearly equal or where ^' — ^ is very nearly 180° ; but, as we shall see hereafter, these cases arc very unfavorable for the determination of the latitude, and are, therefore, always to be avoided in practice If the great circle SS^ passes north of the zenith, we shall have A^ — A negative, or greater than 180°: hence we have also this criterion : we must take q = P — Q or q = P+ Q according as the great circle SS^ imsscs north or south of the zenith. Second Solution.— Bisect the arc SS', Fig. 25, in T; join FT and ZTy and put C = ST=:S'T, D = the declination of T = 90° — PT, I£= the altitude of !r= 90° — ZT, T :^ the hour angle of T = ZPT, F = the angle PTSy Q =-. the angle ZTS, q = the angle FTZ. We have in the triangle PSS' [Sph. Trig. (25)] sin« C = 8in» K(J — S') cos« ) X -]- cos« J (.5 + d') sin* } X BY TWO ALTITUDES. 261 or, adapted for . logarithmic computation, by introducing an auxiliary angle E, sin (7 sin j& = sin i (^ — ^') cos J A | ^308^ sin (7 cos J& = cos i (^ + a') sin M I ^ ^ In the triangle SPT we have the angle PTS = Pj and there- fore in the triangle S'PTwe have the angle PTS' = 180° — P, the cosine of which will be = — cos P: hence, from these triangles we have the equations whence sin D cos + cos D sin C cos P = sin ^ sin J) cos G — cos jD sin (7 cos P = sin d' 2 sin D cos C=Bin d -\- sin d' 2 cos jD sin G cos P = sin ^ — sin d' . « sin i (^ + a') cos H^ — ^0 sm 2> = ^ — ■ ^ ^^ cos C ^ cos i (^ + ^') sin i (^ — ^') cos P = ^^ — ' ^ ^^ cos 2) sin G (309) which determine D and P after Chas been found from (808). In precisely the same manner we derive from the triangles ZTS and ZTS' the equations „ sin Hh + A') cos J (A — h') sm J? = ^^ — ■ — - — - — ^^ cos G ^^ cos i (A + h') sin t (A ~ y) cos H sin (7 (310) Then in the triangle PTZ we have the angle PTZy by the formula q = P-Q and h«nce the equations sin ^ = sin jD sin ff -{- cos D cos ff cos jr cos f cos T = cos jD sin -ET — sin D cos jH" cos q cos ^ sin r = cos Jff sin q 262 LATITUDE. To adapt these for logarithmic computation, let fi and f be deter- mined by the conditions* cos /9 sin ^ = cos H cos q cos /9 cos ^ = sin Jff \ (311) sin fi = cos JS'sin q then f and r are found by the equations sin ^ = cos p sin (D + y) cos 9> cos T = cos /9 cos (D + y) Y (^12) cos ^ sin r = sin ^ To find the hour angles t and t'y let a:=:T — Ki' + O then, since J A = J (<'—<), we have }>l + a: = T — ^ = the angle TPS, iX—x = f —T = the angle TPS\ and from the triangles PTS and PTS' we have sin a X-^x) sin P sin (} A — x) sin P sin C cos d sin (7 cos d' whence sin (} >l -|- a:) — sin (i X — x) cos d' — cos d sin a ^ + x) + sin (J A — x) cos ^' + cos^ and, consequently, tan a; = tan J (^ + d') tan } (^ — d') tan } .1 (818) Hence, finally. As in the first solution, the value of q will become = P+Q when the arc joining the observed places of the stars passes north of the zenith. Example. — 1856 March 5, in the assumed Latitude 39® 17' N. and Longitude 5* 6" 36* W., suppose the following altitudes * The equations (811) can always be satisfied, sinoe the sum of their squares giTet the identical equation 1 = 1. BY TWO ALTITUDES. 268 (already corrected for refraction) to have been obtained; the time being noted by a mean solar chronometer whose daily rate was 10'. 32 lodng. The star Arciurus was not far from the prime vertical east of the meridian ; Spica was near the meridian. Arciurus, h = 18^ 6' 30" Chronometor T = 9* 40« 24'.8 Spica, K = 40 4 43 " T = 14 38 16.7 T^T= 4 57 51.9 d = + 19° 55' 44".6 Corr. for rate = +2 .1 ^' = — 10 24 39 .5 Bed. to aid. time = +48.9 Sid. interval = 4 58 42.9 a = 14» 9- 6'.79 a — a' = 51 29 .1 o! = 13 17 37 .72 k = b 50 12 .0 = 87^33' 0". According to omv first soluiiony we obtain, by (305), B = 91° 15' 52".5 P = 69° 57' 54".7 and, by (306), Q = 64 51 24 .8 wbonco J = 5 6 29 .9 Then, by (307), we find V' = 39°17'20" <' = 5°3'0"= 0*20"12». a' =13 17 37.72 Sidereal time of the observation of Spica = 13 37 49 .72 Sidereal time at mean Greenwich noon =: 22 53 39 .83 14 44 9 .89 Acceleration for 14» 44- 9'.89 = — 2 24.85 " longitude ==— 50.23 Moan time of the observation of Spica = 14 40 54 .81 Chronometer correction at that time, a T' = -|- 2"' 38*.ll According to the second solution^ we prepare the quantities }il=43°46'30" \{d+d')= 4°45'32".6 1(A + A')= 29° 5'36".5 J (5-^') = 15 10 12.1 }(A— A') =— 10 59 6.5 with which we find, by (308), (309), and (310), log tan ^ = 9.437854 D= 6° 34' 32".0 log sin C = 9.854225 P = 68 27 22 .2 log cos C = 9.844639 Q = 108 35 12 .1 log sin I£=^r 9.834176 3 = — 40 7 49 .9 log cos //= 9.863785 26i LATITUDE. (The auxiliaries C and H are not themselves required : we take their cosines from the table, employing the sines as argomenta.) We now find, by (811), (812), (818), and (814), log sin fi == n9.673029 t = 322<> 30' 61".8 log cos fi = 9.946682 x = 1 14 21 .3 r = 39^ 18' 4".0 T — a; == 321 16 30 2) + ;- = 46 52 36 .0 = 21* 26- 6* 9^ = 39 17 20 . iil = 2 65 6 ^ = 18 30 f= 20 12 agreeing precisely with the results of the first solution, 179. In the observation of lunar distances, as we shall see hereafter, the altitudes of the moon and a star are observed at the same instant with the distance of the objects. The ob- served distance is reduced to the true geocentric distance, which is the arc B of the ohove first solution^ or 2 C of the second. The observation of a lunar distance with the altitudes of the objects furnishes, therefore, the data for finding the latitude, the local time, and the longitude. 180. (B.) A fixed star observed at two different times, — ^In this case the declination is the same at both observations, and the general formulse of the preceding articles take much more simple forms. The hour angle k is in this case merely the elapsed sidereal time between the observations, the formula (304), since a = a', becoming here >l = (T'— T) + (aT'— aT) (315) Putting d' for d in (308) and (309), we find JE?= 0, cos P= 0, P= 90°; and Cand D are found by the equations sin C = cos ^ sin } X. sin D = f816) cos a ^ ^ Since we have q=^ P— Q = 90° — §, the last two equations of (311) give sin fi = cos H cos (?, cos y = %m H sec /9 BY TWO ALTITUDES. 265 which, by (310), become* .a COS } (A + A') sin i (A — h') sin p = ^^ — ■ ' ^^ ^ sin G sin i(h + A') cos J (^ — ^') cos y = ^ ' "^ ^ ^ (817) cos fi COS C Then f and r are found by (312), or rather by the following : sin ^ = cos fi sin (D + /) tan/9 8in/9 } (318) tan T = or sin t = ^ cos (D + r) ^^^ 9 The hour angles at the two observations are \Z\~^\\ } (319) Here y^ being determined by its cosine, may be either a posi- tive or a negative angle, and we obtain two values of the latitude by taking either D-\-'( ox J> — 7^ m (318). The first value will be taken when the great circle joining the two positions of the star passes north of the zenith ; the second, when it passes south of the zenith. The solution may be slightly varied by finding D by the formula cos J >l ^ ^ obtained directly from the triangle TTS (Fig. 25), which is right- angled at T when the declinations are equal. We can then dis- pense with C by writing the formulae (317) as follows : . ^ cos J (^ + ^') sin J (A — A') sm /9 = ^^ — ' ^^ ^ cos d sin i A ; (321) sin J (^ + ^') cos } (A — A') sin D cos Y = ^^-^^^ — ' — ^ cos ^ sin b * The formulas (315), (316), and (817) are essentially the same as those first given for this case by M. Caillet in his Manuel du Navi^atewr^ Nantes, 1818. 266 LATITUDE. 181. (C.) The sun observed at (wo different times. — ^In this case the hour angle X is the elapsed apparent solar time. If then the times T and T' are observed by a mean solar chronometer, and the equation of time at the two obser\'ations is e and e' (positive when additive to apparent time), we have >l = (T'— T) + (Ar— Ar) — (f'— (822) Taking then the declinations d and 5' for the two times of obser- vation, we can proceed by the general methods of Art. 178. But, as the declinations differ very little, we can assume their mean — i.e. the declination at the middle instant between thc^ observations — as a constant declination, and compute at least an approximate value of the latitude by the simple formulae for a fixed star in the preceding article. We can, however, readily correct the resulting latitude for the error of this assumption. To obtain the correction, we recur to the rigorous fomiulie of our second solution in Art. 178. The change of the sun's declination being never greater than 1' per hour, we may put cos |(J — 3^ = 1. Making this substitution in (308) and (309), and putting 8 for J (5 — o') so that d will signify the mean of the declinations^ and A J for J (d' — d) so that sS will signify one-half the increase of the sun's declination from the first to the second observation (positive when the sun is moving northward), we shall have aJ = — H^ — ^') sin A^ tan ji&= — cos o tan } A But Au diminishes with L so that £ always remains a small quantity of the same onler as ao ; and hence we may also put cos jET— 1. Thus the se<.'v>nd equation of ^oOS) gives siu C = cos o sin } i and the first of \oOi>) . sin a smi)= which are the same as i-^l*»u as iri^en for the case where the V declination i< ab>^>Iutolv invuriaWc. Ilonce :vo sensible error i* prvHiiU'cd in the values of (.'and D by the us<' of the mean de- BY TWO ALTITUDES. 267 clination. But by the second equation of (309) P will no longer be exactly 90° : if then we put P = go** + aP we have, by that equation, cos dmn^d sin Ad sin aP = cos D sin C cos D sin i il or simply cos D %\xi\k Then, since q^=P — §, we have 3 = 90O — C + aP The rigorous formula for the latitude is sin ^ = sin D sin H -|- cos D cos Jff cos q in which when we use the mean declination we take q = 90° — Q: therefore, to find the correction of ^ corresponding to a cor- rection of y = aP, we have by differentiating this equation, D and jBT being invariable, cos ^.A^ = — COS D COS JS'sin q,AP Ad cos H cos Q sin } X We have found in the preceding article sin fi = cos JBT cos Q; and hence A^ = ^l^ILL^ (323) COS ^ sm } A In the case of the sun, therefore, we conapute the approximate ktitude f by the formula (316), (317), and (318), employing for 8 the mean declination. We then find Ay> by (323) (which in- volves very little additional labor, since the logarithms of sin /9 and sin J A have already occurred in the previous computation), and then we have the true latitude If we wish also to correct the hour angle r found by this method, we have, from the second equation of (47) applied to 268 LATITUDB. the triangle PTZ (taking b and c to denote the sides FT and Z Tj which are here constant), cos H cos A At = .aP cos <p in which A is the azimuth of the point T. But we have in the triangle PTZ cos H cos A = sin f> cos D cos t — cos ^ sin Z> Substituting this and the value of aP, we have A^ -rv At = (tan cos T — tan D) sinU ^ -^ and, substituting the value of tan D (320), At A<J / , tan ^ \ = I tan f cos T I sin i A \ cos \ki When this correction is added to r, we have the value that wonld be found by the rigorous formulae, and we have then to apply also the correction z according to (814). In the present case we have, by (313), re = — A^ tan d tan } X and the complete formulae for the hour angles / and V become t =t-|-at — X — \k f = T + Ar — X + JA Putting y = At — X we find, by substituting the above values of at and x, ^^^ /tan^^C08r__tan£\ " \ iiti\k tan J -I ' and then wc have t =T+y— U <'=f + y + J^ I (825) The corrections Ay> and y are computed with sufficient accu- racy with four-place logarithms, and, therefore, add but little to the labor of the computation. Nevertheless, when both latitude and time are required with the greatest j)reei8ion, it will be pre- ferable to compute by the rigorous fonnulre. BY TWO ALTITUDES. 269 Example.— 1856 March 10, in Lat. 24^ K, Long. 80^ W., suppose we obtain two altitudes of the sun as follows, all correc- tions being applied : find the latitude. At app. time 0* 30" h = 61^ 11' 38";3 (^ ) = — 3^ 61' 62".8 « 4 30 A' = 18 46 35 .8 (3')= — 3 47 57 .4 iX= 2* 0" i(h + h') = S9 59 7.1 ^=— 3~"49 55 .1 = 30<> 0' * (A — A') = 21 12 31 .3 a^ = + 1' 57".7 The following is the form of computation by the formulie (316), (317), and (318), employed by Bowditch in his Practical Navigator^ the reciprocals of the equations (316) and of the second of (317) being used to avoid taking arithmetical comple- ments. This form is convenient when the tables give the secants and cosecants, as is usual in nautical works. eoseo i X 0.801080 sec 6 0.000972 coseo nl.l76024 cosec C 0.802002 cos 9.987854 cos 9.987864 cos } (* + *') «.884847 cosec 0.192066 2> = — 4o 26' 21^.8 cosec fil.112878 sin J (A — A') 9.668428 sec 0.080469 rini? 9.744777 cos 9.919829 cos 9.919829 sec 0.080207 y= 88 46 88 .0 /> + y == 29 20 16 .7 sin 9.690161 = 24® 2' 28".2 sin 9.609990 K the approximate latitude had not been given, we might also have taken D — r = — 38° 10' 69''.3, and then we should have had C08/9 9.919829 sin (P — r) n9.791113 f^ = — 30^ 55' 44".3 sin ip n9.7To942 To correct the first value of the latitude for the change of declination, we have, by (323), log A^ 2.0708 sin p 9.7448 cosec } X 0.3010 Becs«> 0.0394 Af = — 143".2 log Ckf n2.1560 and hence the true latitude is SP' = 24^' 2' 23".2 — 2' 28".2 = 24o 0' 0" 270 LATITUDE. which agrees exactly with the value compnted by the rigorond formula. The approximate time is found by the last equation of (318) with but one new logarithm : we have Bin /9 9.744777 cosf' 9.960596 T = 37° 28' 23" Bin r 9.784181 By (824) and (325), we find log A^ 2.0708 log A^ 2.0708 cosec i X 0.3010 cot i X 0.2386 tan ^ 9.6494 tan d yi8.8259 cos T 9.8996 — 13".7 nl.l353 + 83".3 1.9208 y = + 83".3 — (— 13".7) = + 97" T + y = 37°30' 0" = 2»30-0' t= 0* 30* 0* r = 4* 30« 0- which are perfectly exact. . 182. (D.) Tico equal altitudes of the same siar^ or of the sitn. — ^This case is a verj' useful one in practice with the sextant when equal altitudes have been taken for determining the time by tlie method of Art. 140. By putting /i' = A in the formulae (817), we find sin ^ = 0, cos /3 =^ 1, and hence (318) gives sin ^ = sin (D + y\ or f=D + Y. We have, therefore, for this case, by (320) and (321), _ tan d sin h sin D tan 2) := cos y = ; cos i X sin d i (qo^\ which is the method of Art. 164 applied as proposed in Art. 165. We give y the double sign, and obtahi two values of the latitude, for the reasons already stated. The time will be most conveniently found by Art. 140. The method there given is, however, embraced in the solution of the present problem. For, since we here have sin ^ = 0, we also have r = 0, and the hour angles given by (325) become t =y-iX f = y + iX BY TWO ALTITUDES. .271 the mean of which giveg i(t + f) — y = that is, — y is the correction of the mean of the times of obser- vation to obtain the time of apparent noon = 0*. This correction was denoted in Art. 140 hy ^T^; and since cos r = 0, the formula (324) gives, when divided by 15 to reduce it to seconds of time, ^ Ad tan ^ Ad tan d 15 sin iX 15 tan i X which is identical with (262). Thus it appears that (262) is but a particular case of the formula (324), which I suppose to be new. The latitude found by (326) will require no correction, since sin ^ = 0, and therefore A(p = 0, NoTK. — The preceding solutions of the problem of finding the latitude and time by two altitudes leare nothing to be desired on the score of completeness and accuracy ; but some brief approximatire methods, used only by navigators, will be treated of among the methods of finding the latitude at sea, and in Chapter VIII. 183. I proceed to discuss the effect of errors in the data upon the latitude and time determined by two altitudes, and hence also the conditions most favorable to accuracy. Errors of altitude. — Since the hour angles t and <' are connected by the relation <' = < + ^j the errors of altitude produce the same errors in both ; for, k being correct, we have dt^ = dt ; and for either of these we may also put rfr, since we have, in the second general solution of Art. 178, T — x = i{t + <'), and x is not affected by errors of altitude. Now, we have the general relations sin h = sin ^ sin d -\- cos f cos d cos t 1 /q97\ sin A' = sin ^ sin ^' -f cos ^ cos ^'cos t' ) which, by differentiation relatively to A, ^ f , and ij give [see equations (51)] rf A = — cos Ad^ — cos ^ sin A dr dh' = — cos A'df — cos f sin A'dr in which A and J.' denote the azimuths of the two stars, or of the same star at the two observations. 272 LATITUDE. Eliminating dz and dtp successively, we find , sin A' ,, . sin A ,,, ^ sin (A! — A) ^ sin (A' — A) . cos A' ,- cos J. ... cos (pdr = -: dh ; dh' ^ sin (A' — A) sin (A' — A) (828) These equations show that, in order to reduce the effect of errors of altitude as much as possible, we must make sin {A' — A) ss great as possible, and hence A' — A, the difference of the azi- muths, should be as nearly 90° as possible. If we suppose A' — A = 90°, we have d^ = — sin A'dh + smAdh' cos <pdr = cos A' dh — co& A dh' and, under the same supposition, if one of the altitudes is near the meridian the other will be near the prime vertical. If, then, the altitude h is near the meridian, sin A will be small while sin A' is nearly unity, and the error d^ will depend chiefly on the term sin A'dh. At the same time, cos A will be nearly unity and cos A' small, so that the error rfr will depend chiefly on the term co& A dh^. Hence the accuracy of the resulting latitude will depend chiefly upon that of the altitude near the meridian ; and the accuracy of the time chiefly upon that of the altitude near the prime vertical. If the observations are taken upon the same star observed at equal distances from the meridian, we have A' = — Aj and the general expressions (328) become cos ^c?r = — 2 cos A dh — dh' 2 sin A The most favorable condition for determining both latitude and time from equal altitudes is sin A = cos A^ or A = 45°. Errors in the observed clock times. — An error in the observed time may be resolved into an error of altitude ; for if we say that dr is the error of T at the obser\^ation of the altitude A, it BY TWO ALTITUDES. 278 amounts to saying either that the time T — rfjP corresponds to the altitude A, or that T corresponds to the altitude h + rfA, dh being the increase of altitude in the interval dT. We maj^ therefore, consider the time T as correctly observed while h is in error by the quantity — dh. Conversely, we may assume that the altitudes are correct while the times are erroneous. The discussion of the errors under the latter form, while it can lead to no new results, is, nevertheless, sufficiently interesting. We have, from the formula (804), dX = dT'-'dT The general equations (827), upon the supposition that h and h' are correct, give = — cos Ad^ — cos ^ Bin Adt = — cos A'df — cos f sin A' (dt + dX) where we put dt + dX for di\ since t' = i + L Eliminating dty we find cos sin A' sin A „ ^^^^^ d0 = dk (829) ^ 8in(^'— ^) ^ ^ Eliminating df^ and similarly -, sin A' cos A ,, or = ; dl sin {A' — A) -^ sin A cos A' ,, (BT = dX sin {A* — A) But we have from the formula r — x = J (< + <') dr=\(dt^ dV) and hence _^sin(^-+^) jj. 8in(J.'— ^) 2 ^ ^ K we denote the clock correction at the time T by #, we shall have and d^ = dt'^ dT Vol. L— 18 274 LATITUDE. or, if we deduce ?? from the second observation, the rate being supposed correct, The mean is d^ = dT — i(dT+dT') Substituting the value of rfr and also dX = dT' — dT^ we find, after reduction, ^^^ sin^cosA- ^y^ ain^-cos^ ^y, 8in(J.' — il) 8in(A'— ^) ^ ^ The conclusions above obtained as to the conditions favorable to the accurate determination of either the latitude or the time are, evidently, confirmed by the equations (329) and (331). In addi- tion, we may observe that if dT' = dT^ we have dip = and d& =^ dT: a constant error in noting the clock time produces no error in the latitude, but aftects the clock correction by its whole amount Errors of declination. — These errors may also be resolved into errors of altitude. By difterentiating the equations (327) rela- tively to h and 5, we find dk = cos qd9, dh' = cos ^dd' in which q and 7' arc the parallactic angles at the two observa- tions' respectively. If these values arc substituted in (328), the resulting values of dip and dr will be the corrections required in tlie latitude and hour angle for the errors dd and dd' ;* and, de- noting these corrections by a^ and Ar, we have ein-4'co8<7 ,^ . sin il coso^ ,^, ^ 8in(ii' — ^) ^8ih(.l'— ^) cos A* cos a , ^ cos A cos q' , ,. cos cp AT = do ^— dd' ^ sin {A' — A) sin (^' — A) (832) If the observation h is on the meridian, and A' on the prime vertical, we have a^ = — dd ; and in the same case we have, by * The error of a quantity and the correction for this error are too frequently cob- founded. They arc numerically equal, but have opposite signs. If a thooM h% a — 7, it is too great by z; its error i% -\- x; but the correction to reduce it to its true value is — x. BY TWO ALTITUDES, 275 (828), df=^+ dhy and the total correction of the latitude = rfA — dd^ precisely the same as if the meridian observation were the only one. Hence there is no advantage in combining an observation on the meridian with another remote from it, iii the determination of latitude: a simple meridian observation, or, still better, a series of circummeridian observations, is always preferable if the time is approximately known. When the sun is observed and the mean declination is em- ployed, putting A^ = J {d' — d)y we have dd = a5, dd^ = — a^, and, by (832), sin A' cos ^ -|- sin A cos q' 8in(A'— il) which, by substituting . ,, sin gr cos d . . sinq' cosd sm A' = — ^ sm A = — cos f cos ^ becomes ^^___Bin^+|^ ^^ (333) sm (A — A) cos ^ This is but another expression of the correction (323). If, when the sun is observed, instead of employing the mean declination we employ the declination belonging to the greater altitude, which we may suppose to be A, we shall have dd = 0, d8^ = — 2 a5 ; and, denoting the correction of the latitude in this case by a'^ , we have, by (382), , 2 sin A cos a* ^ 2 »\nq cos (/ cos d aV = ^ . A^ = ^ ^ A^ sin (A' — A) sin (A' — A) cos ^ Under what conditions will a'^ be numerically less than Af ? First If both observations are on the same side of the meridian, the condition a'^ < Af gives 2 sin q cos ^ < sin (j' -|- q) or 2 sin q cos ^r' < sin ^ cos q + cos ^ sin q whence tan q < tan ^ which condition is always satisfied when h is the greater altitude. Secondly. If the observations are on different sides of the 276 LATITUDE. meridian, q and q' will have opposite signs, and we shall have, numerically, sin (9' — q) instead of sin (9' + q). The condition A'f < ^f J then, requires that 2 sin q cos 5^ < sin 5'' cos q — cos ^ sin q or tan J < I tan 5'' Therefore a <p will be greater than Ay> on/y when the observa- tions are on opposite sides of the meridian and tan y > i tan <^. In cases where an approximate result suffices, as at sea, and the correction a^ is omitted to save computation, it will be expedient to employ the declination at the greater altitude, except in the single case just mentioned.* But to distinguish this ease with accuracy we should have to compute the angles q and q' ; and therefore an approximate criterion must suffice. Since the parallactic angles increase with the hour angles, we may substi- tute for the condition tan g > J tan q' the more simple one < > J <', which gives ^ 2 or {t and i' being only the numerical values of the hour angles) Hence we derive this very simple practical rule : employ the sun's declination at the greater altitude, ercejH xchen the time of this altitude is farther from noon than the middle time, in which case employ the mean declination. The greatest error committed under this rule is (nearly) the value of A^ in (323), when t ^ t. But since in this case 3/ - /', and t -\- t^ = Xy we have r — ^ ^, and therefore sin /S := cos f sin r = cos (f sin } ?., This reduces (323) to a^ - - — J a5 sec } i. Since X will seldom exceed 6*, Ad will not exceed 3', and the maxinnuu error will not exceed I'.G. In most cases the error will be under 1', a degree of approximation usually quite suffi- cient at sea. Nevertheless, the computation of the correction A^ by our formula (323) is so simple that the careful navigator * BowDiTCii and navigators generally employ in all cases the mean declination; hut the ahoTe discussion proves that, if the cases are not to be distinguished, it will be better always to employ the declination at the greater altitude. BY TWO EQUAL ALTITUDES. 277 will prefer to employ the mean declination and to obtain the exact result by applying this correction in all cases. SIXTH METHOD. — BY TWO ALTITUDES OF THE SAME OR DIFFERENT STARS, WITH THE DIFFERENCE OF THEIR AZIMUTHS. 184. Instead of noting the times corresponding to the observed altitudes, we may observe the azimuths, both altitude and azi- muth being obtained at once by the Altitude and Azimuth Instrument or the Universal Instrument. The instrument gives the difference of azimuths = X. The computation of the latitude and the absolute azimuth A of one of the stars may then be performed by the formulre of the preceding method, only inter- changing altitudes and declinations and putting 180° — A for U When A has been found, we may also find t by the usual methods. SEVENTH METHOD. — BY TWO DIFFERENT STARS OBSERVED AT THE SAME ALTITUDE WHEN THE TIME IS GIVEN. 185. By this method the latitude is found from the declinations of the two stars and their hour angles deduced from the times of observation, without employing tlw altitude itself, so that the result is free from constant errors (of graduation, &c.) of the instrument with which the altitude is observed. Let 8, 0' = the sidoroal times of the observations, a, a' = the right ascensions of the stars, df d' = the declinations ** " t, f = the hour angles " " h = the altitude of either star, ^ = the latitude ; then, the hour angles being found by the equations we have sin A = sin ^ sin d -{- cos ^ cos d cos t sin h = Bin ip sin d' -f- cos ^ cos d' cos f Prom the difference of these we deduce tan <p (sin d' — sin d) = cos d cos t — cos ^' cos f = cos a cos t — cos d' cos r = ^ (cos ^ — cos d') (cos t + cos f) + ] (cos d + cos d') (cos t — cos f) 278 LATITUDE. and, by resolving the quantities in parentheses into their fiEtctors, tan ^ = tan i (^' + d) cos i(f + t) cos } (f — f) + cot i Id' — d) sin i(f + f) sin iif — f) }(3a4) If now we put m sin M= sin i(f — t) cot } (^' — a) m cos Jlf = cos i(f — t) tan } (d' -{- S) we have tan ^ = m cos [} (^ + f) ^ 3f] (835) The equations (334) determine m and JHf, and then the latitude is found by (335) in a very simple manner. It is important to determine the conditions which must govern the selection of the stars and the time at which they are to be observed. For this purpose we differentiate the above expres- sions for sin A relatively to tp and /. The error in the hour angles is composed of the error of observation and the error of the clock correction. If we put Tj T' = the observed (sidereal) clock time, A T = the clock correction, dT = the rate of the clock in a unit of clock time, we shall have t = T-\- ^T^a, f = T + £iT + dT{T — T) — a' Difterentiating these, assuming that the rate of the clock is suffi- ciently well known, we have dt=^dT+d£iT, df = dT + d£,T in which rfJ", dT' arc the errors in the observed times, and (/a T the error in the clock correction. The difterential equations arc then dh = — cos A dtp — cos if 9\n AdT — cos tp B\ii A d^T dh = — cos A' dip — cos <p sin A' dT' — cos <p ^\n A' d c^T in which A and A' arc the azimuths of the stars. The difference of these equations gives -'If- = «!?_^ dT-v ?i^^^ d r + gi"_ji: - ^'" ^ rf ^T cos^ COS -4— COS A' cos -4 — cos -A' cos^' — cos^ BY TWO BQUAL ALTITUDES. 279 The denominator cos A — cos A' is a maximum when one of the azimuths is zero and the other 180°, that is, when one of the stars is south and the other north of the observer. To satisfy this condition as nearly as possible, two stars are to be selected the mean of whose declinations is nearly equal to the latitude, and the common altitude at which they are to be observed will be equal to or but little less than the meridian altitude of the star which culminates farthest from the zenith. It is desirable, also, that the difference of right ascensions should not be great. The coefficient of dcs.T\% equal to — cot \{A! + -4), which is zero when \{A* + A) is 90° or 270° : hence, when the observa- tions are equally distant from the prime vertical, one north and the other south, small errors in the clock correction have no sensible effect. When the latitude has been found by this method, we may determine the whole error of the instrument with which the altitude is observed; for either of the fundamental equations will give the true altitude, which increased by the refraction will be that which a perfect instrument would give. 186. With the zenith telescope (see Vol. 11.) the two stars may be observed at nearly the same zenith distance, the small difference of zenith distance being determined by the level and the micrometer. The preceding method may still be used by correcting the time of one of the observations. If at the ob- served times T, T' the zenith distances are ^ and ^', the times when the same altitudes would be observed are either C — C T and r' + cos ip sin A! or, T-\ — and T cos ip sin A where f ' — ^ is given directly by the instrument. With the hour angles deduced from these times we may then proceed by (834) and (335). But it will be still better to compute an approximate latitude by the formulae (334) and (335), employing the actually observed times, and then to correct this latitude for the difference of zenith distance. 280 LATITUDE. By diiFerentiating the formula tan f (sin ^' — sin S) = cos ^ cos t — cos ^' cos t relatively to tp and i\ we have sec' ip (sin ^' — sin ^) dip = cos ^' sin ^ d^ = sin C flin A' cW Ilenee, substituting dje = dT = c-c COS ^ sin A' we find df = - } (C — C) sin C cos ip sin 1 (a' — a) cos 1 (^' + S) and the true latitude will be ^ + dtp. (836) EIGHTH METHOD. — BT THREE OR MORE DIFFERENT STARS OBSERVED AT THE SAME ALTITUDE WHEN THE TIME IS NOT GIVEN. 187. To find both the latitude and the clock correction from the dock times when three different stars arrive at the same altitude. As in the preceding method, we do not employ the common altitude itself; and, as we have one more observation, we can de- termine the time as well as the latitude. Let 5, aS", S'% Fig. 26, be the three points of the celestial sphere, equidistant from the zenith Z, at which the stars are observed. Let Fig. 26. Also, let T, T\ T" = the clock times of the observations, ^T = the clock correction to sidereal time at the time T, dT = the rate of the clock in a unit of clock time, o, a', a" =- the right ascensions of the stars, dy d\ d*' = the declinations fj fy f' ^r the hour angles h = the altitude, tp = the latitude. X =f ^t = SPS', a it ti (( then, since the sidereal times of the observations are BT THREE EQUAL ALTITUDES. 281 e = T + aT S' = T' + £iT + dT (T' —T) and the hour angles are we have X = r— T+dT(r— T) — (a' —a) r = T"— T + dxlT'— T) — (a" — a) The angles X and X' are thus found jfrom the observed clock times, the clock rate, and the right ascensions of the stars. The hour angles of the stars being ^ < + A, and t + A', we have sin A = sin ^ sin d -)- cos ^ cos d cos t sin A = sin ^ sin d' -f ^^^ 9 cos d' cos (t + X) sin A = sin «p sin d" -\- cos ^ cos ^" cos (^ + ^') ^ sin ^" + COS sp COS ^" cos (^ + ^^ Subtracting the first of these from the second, we have an equa- tion of the same form as that treated in Art. 185, only here we have i + X instead of i' ; and hence, by (334), we have m sin Jf = sin i X cot } (^' — ^ 1 r337^ m cos M = cos } X tan } (^' + ^) / and putting N=::iX-^M (338) we have, by (335), tan ^ == m cos (t + N) (339) In the same manner, from the first and third observations we have m' sin Jf' = sin i X' cot i (^" -— d) w! cos M* = cos J A' tan i (5" + d) N'=\X' — M' (341) tan sp = m' cos (^ + N') ' (342) I (340) The problem is then reduced to the solution of the two equa- tions (339) and (342), involving the two unknown quantities f and /. K we put k cos (^ + iV) = — 282 LATITUDE. there follows also kcoB(t + iV') = -^ and these two equations are of the form treated of in PL Trig. Art. 179, according to which, if the auxiliary i> is determined by the condition tan * = -, (343) m we shall have tan p + } (N+ iVT')] = tan (45*» — ^) cot i(N'—N) (844) which determines ^, from which the clock correction is found by the formula The latitude is then found from either (339) or (342).* To determine the conditions which shall govern the selection of the stars, we have, as in Art. 185, - cos A d^ — cos 9> sin A dT — cos ^ sin ji d^T • cos A' dip — cos f sin A' dT' — cos ^ sin A' dAT dh = — cos A'^d^ — cos ^ sin A"dT" — cos ^ sin A^d^T By the elimination of d^T^ we deduce the following: (sin A — sin A')dhz=^ sin (^' — ^ ) d^ — cos ^ sin A sin A (dT — dT ) (sin A' — sin A") dh = sin \a" — A' ) d^ — cos ^ sin A" sin A' (dT*' — dV ) (sin A" — sin ^ ) (/A = sin {A — A") d^ — cos sin A sin A'* \dT — dT*) Adding these three equations together, and putting 2 iT = sin {A' — A)-\- sin {A" — A') + sin (^ — A") dh = dh = we find d6 sin A (sin A" — sin A') sin A' (sin A — sin A") ^ ^ ' dT-\ ^ dr cos^ 2K 2K sin A" (sin A' — sin ^) ,_,, 2K By eliminating d<p from the same three equations, we shall find * This simple and elegant solution is due to Gacss, Monatlicht Corre*pondenz^ VoL XVIII. p. 287. BY THREE EQUAL ALTITUDES. 283 d^T = »^P ^ (<^Q« ^' — CQ8 "^ 1 ^jf I gin -4^ (cos A" — cos A) ^^, 2K IK sin A" (cos -<4 — cos A') ,_,, The denominator 2^ is a maximum when the three differences of azimuth are each 120°,* which is therefore the most favorable condition for determining both the latitude and the time. In general, small differences of azimuth are to be avoided. Gauss adds the following important practical remarks. It is clear that stars whose altitude varies slowly are quite as available as those which rise or fall rapidly ; for the essential condition is not so much that the precise instant when the star reaches a supposed place should be noted, as that at the time which is noted the star should not be sensibly distant from that place. We may, therefore, without scruple select one of the stars near its culmination, or the Pole star at any time, and we can then easily satisfy the above condition. Moreover, at least one of the stars will always change its altitude rapidly when the condition of widely different azimuths is satisfied. The stars proper to be observed may be readily selected with the aid of an artificial globe, and in general so that the intervals of time between the observations shall be so small that irregu- larities of the clock or an error in the assumed rate shall not have any sensible influence. Having selected the stars, the clock times when they severally arrive at the assumed altitude are to be computed in advance within a minute or two, so that the observer may be ready for each observation at the proper time. This is quickly done with four-place logarithms by the formula (267), in which (p and ^ will have the same values for the three stars. • For by patting a=. A' — J, a' = A" — A'^ we have 2 JST = sin 11 -}- sin a! — sin (a + a') and, differentiating with reference to a and a\ the conditions of maximum or mini- mam are cos a — cos (a -f" <*')== ^ cos a' — cos (a -|- a') = which give either a :— a' :^ or a = a' = 120®; and the latter evidently belongs to the case of maximum. 284 LATITUDE. If it is desired to compute the differential formulffi, the follow- ing form will be convenient We have JT = — 2 sin } (^' — A.) sin } (A" — A') sin 1 (A — A") d^ __ sin A cos } (A" + A') sin j (A" ^ A') 15 cos ^ K sin A' cos iU + ^'0 sin i(A — A") ^ . sin^cos } (A' + ^) sin } (A' - ^) .^, + ^ dT , Bin ^' sin ijA + A") sin } (^ — X") _, Bin A" Bin i (X' + A) Bin | (A> — .1) ^^„ where dip is divided by 15, since it will be expressed in seconds of arc, while dT^ dT\ and dT^' are in seconds of time. If we first compute the coefficients of the value of d^T^ those of df will be found by multiplying the former respectively by cot J {A' + A'')y cot l(A + A'% and cot J {A' + A), and also by 15 cos f. It is well to remark, also, for the purpose of verifica- tion, that the sum of the three coefficients in the formula for df must be — 0, and the sum of those in the formula for rf^T^must be = - 1. The substitution of dX for dT-dT, and dX' for dT'—dT, will reduce the above expressions to a more simple form, which I leave to the reader. Example. — To illustrate the above method. Gauss took the following observations, with a sextant and mercurial horizon, at Gottingen, August 27, 1808. The double altitude on the sextant was 105° 18' 55". The time was noted by a sidereal clock whose rate was so small as not to require notice. BT THREE EQUAL ALTITUDES. 285 a Andromeda T == 21* 88" 26* a Urs(B Minoris T = 21 47 80 aiyroj 2"= 22 6 21 The apparent places of the stars were as follows : a AndromedfB a a TJtscr Minoris a! a Lyr<B n 28» 58- 33'.83 55 4.70 18 30 28.96 d =28*» 2'14".8 d' = 88 17 5 .7 ^" = 38 87 6 .6 Hence we find j;i = — 5**18'25".28 \ (d' — d)= 30 7 25 .45 i (d' + d)= 58 9 40 .25 log cot } (^' — ^) 0.2363973 log sin i X n8.9661070 log msinM n9.2025043 log tan } (^' + d) 0.2069831 log cos U 9.9981343 log m cos if 0.2050674 } A' = 44<> 59' 55".28 i (a" — ^) = 5 17 25 .90 i (^" + ^) = 88 19 40 .70 log cot } (^" — d) 1.0333869 log sin i X' 9.8494751 log m' sin M' 0.8828620 log tan 1 (r + d) 9.8179461 log cos } X' 9.8494949 log m' cos JBT 9.6674410 log tan M log cos M log m n8.9974369 9.9978645 0.2072029 log tan M' log sin M* \ogm' Jf == — 5^0^ 87".96 W = Ji— Jf = J\r= + 22 12.68 hX'--M'=ir= 1.2154210 9.9991963 0.8836657 86*» 3^ 55".07 41 30 59 .79 m * = 11** 58' 41".28 log ^ = log tan « 9.3235372 450 _^ ,3= 33 6 18 .72 log tan (45*» — *) 9.8142617 \{N'—N) = — 20 56 36 .24 log cot J (JV^ — JV) n0.4171063 t+}(JV^' + JNr) = — 59 35 14 .71 logtanp+}(-ZV''+^^]n0.2313680 K^ + ^ = — 20 84 23 .56 t= — 39 51.15 = — 2»36- 3*.41 a= 23 58 83.33 ^ + a = e = 21 22 29 .92 T= 21 33 26. Clock correction a T = — 10 56 .08 Then, to find the latitude, we have 286 LATITUDE. t + N= — 88^ 88' 38".47 / _[. JVT = — 80*» 31' 50".94 log cos (t + N) 9.8926738 log cos {t + N*) 9.2162110 logm 0.2072029 log m' 0.8836667 log tan ^ 0.0998767 log tan ^ 0.0998767 q> = 51*^ 31' 5r'.46 If with these results we compute the true altitude of the stars, we find from each h = 52° 37' 21".2. The refraction was 42".7, and hence the apparent altitude = 52° 38' 3".9. The double altitude observed was, therefore, 105° 16' 7".8. The index correction of the sextant was — 3' 30", and hence the double altitude given by the instrument was 105° 15' 25", which was, consequently, too small by 43". To compute the differential equations, we find A = 293*^ 45'.2 A' = 182*^ 9'.1 A" = 90° 17'.9 and hence dip = ^ 3.808 dT— 0.288 <f T' — 3.519 dT' db.T= — 0.391 (fT— 0.007 dT — 0.602 dT" by which we see that an error of one second in each of the times would produce at the most but 7".6 error in the latitude, and one second in the clock correction. 188. Solution of the preceding problem by Cagnoli's fomiidce. — After Gauss had published the solution above given, he was himself the first to observe* that Cagnoli's formulse for the solution of a very difierent problemf might be applied directly to this. Wlien the altitude is also computed, Cagnoli's formulie have slightly the advantage over those of Oauss. To Pig. 2«. (6i«).^ deduce them, let 7, 7', 7" be the parallactic angles at the three stars, or (Fig. 26) let qr=PSZy q' = PS'Z, q'' = PS"Z, and also put q =iiPS"S' — PS'S") Q' = i iPS"S — PSS") Q" = i(PS'S —PSS') * Monatliche Corretpondenz, Vol. XIX. p. 87. t Namely, that of determining, from three heliocentric places of a solar spot, tht position of the sun's equator, and the declination of the spot. — See Caoxou*s Trigonomitriey p. 488. BT THREK EQUAL ALTITUDES. 287 then, aince 2iSS', ZS'S", and ZSS " are isosceles triangles, we have q + PSS' = PS'S — 4 ^ + PS'S" = P8"S' — g" q + PSS" = PS"S — g" whence ? + 2' = 2$" «'+?" = 2(2 S" + g =2$' « + 3' + «"=«+ <2'+ «" i =-Q+Q'+Q" ) 2'= C -«'+$" k345) «" = «+«'-«" J Now, Q, $', Q" are found from the triangles P<S'"<S", PS"S, and PS'S, by Napier's Analogies (Sph. Trig. Art. 73), thus : tan Q = i^ icot } (X' — l) C08l(a"+i') ^ ^ ^ ^, Bin i id" — d) ^ , ,, . tanr= '^"*^''~'^ cotM COS i (a' + S) where ^, ^' are the angles at the pole found as in the preceding article. With these values of §, §', §", those of y, g', and j" become known by (845). We have also whence and firom this or cos ^ sin (f '\- X^ = cos h sin ^f' cos ^ sin ^ = cos h sin q sin (^ + ^) sin q[ sin f sin q sin (^ + >l) + s5^ ^ sin ^ + ^i^ ? sin (f + >l) — sin f sin g' — sin q tan (e + U) _ tan K^ + g) tan 1 A tan } (^ — s^) 288 LATITUDE. Substituting the values of q and q' in terms of §, this ^ves tan (t + iA) = tan ik tan Q" cot (Q — Q') or, substituting the value of tan ^', tan (< + U) = "" I ^^1 7 2 cot ( e - C) (347) COS } (^ + V which determines < + J ^, whence i and the clock correction. "We can now find the latitude and altitude from any one of the triangles PSZ, PS'Z, PS^'Z, by Napier's Analogies (Sph. Trig. Art. 80) : thus, from PSZ we have tan KsP + K) = ^^iLOHh?) ^^ ,4^0 ^ j^) cos i (^ — q) tani(^ -h) = ^'^\^l ^ |cot (450 + H) sin i (^ + ^) andthen^ = J(f + A) + i(f-'i)» A = H?^ + 'i)-J(f-A). . As all the angles are determined by their tangents, an am- biguity exists as to the semicircle in which they are to be taken; but, as Gauss remarks, we may choose arbitrarily (taking, for example, (>, (?', Q'^ always less than 90°, positive or negative according to the signs of their tangents), and then, according to the results, will have in some cases to make the following changes : 1. K the values of tp and h found by (348) are such that cos ip and sin h have opposite signs, we must substitute 180° -r q for q and repeat the computation of these two equa- tions. In this repetition the same logarithms will occur as before, but difterently placed. 2. If the values of f and h exceed 90°, we must take their supplenieuts to the next multiple of 180°. 3. The latitude is to be taken as north or south according as sin <p and sin Ii have the same or different signs. No ambiguity, however, exists in practice as to /+ \K found by (347), since Q — §' can differ from its true value only by 180°, and this difference does not change the sign of cot (§ — Q^)i hence tan (t + JA) will come out with its true sign; and between BY THREE OR MORE EQUAL ALTITUDES. 289 the two values of ^+ JJl, differing by 180°, or 12*, the observer will be at no loss to choose, as he cannot be uncertain of his time by 12*. Example. — Taking the example of the preceding article, we shall find ^ == — 37*» 57' 9".3 C' = + 6*» 17' 5r.66 Q"= ~ 84<> 25' 23".81 J = — C + e' + C" = ~ 40*» 10' 22".85 e = — 39 51 .27 i(t + g) = — SG** 35' 37".06 i(t --q) = + O*' 34' 45".79 i (V + h) = 52 4 36 .35 i (sp — A) = — 32 44 .84 ^ = 51 31 51 .5 h = 52 37 21 .2 189. If we have observed more than three stars at the same altitude, we have more than sufficient data for the determination of the latitude ; but by combining all the observations we may obtain a more accurate result than from only three. This combination is effected by the method of least squares, according to which we assume approximate values of the unknown quantities and then determine the most probable corrections of these values, or those which best satisfy all the observations. Let T, T\ T\ T", &c. be the observed times by the clock when the several stars reach the same altitude. Let a The the assumed clock correction at some assumed epoch = Tq\ 8T\h^ known rate. Let <p and h be the assumed approximate values of the latitude and altitude. With <p and A, which will be the same for all the stars, and with the declinations 5, 5', 5", &c., compute the hour angles <, t\ <", &c. and the azimuths -4, A', -4", &c. If the assumed values were all correct and the observations perfect, we should have a + t= T+ £,T-\' dT{T— T^y.iov each of these quantities then represents the sidereal time of observation ; but if <p, A, and a T require the corrections dtp^ dh, and cJaT', and if dt is the corresponding correction of <, we shall have a + ^ + jt= r+ AT + rfAT + arcr— To) The relation between dip, dh, and dt is dh = — cos Adif — 15 cos <p sin Adt and a similar equation of condition exists for each star. In all Vol. L— 1» 290 LATITUDB. these equations, dh and df are the same, but dt is different for each. K we put f = T + £iT+dT(^T — r.) — (a +t) which are all known quantities, we have dt=f+d£iT, dt'=f + dAT,&c. and the equations of condition become dh-{-coBA .d^ -{-15co8^BmA .^AT-[-15co6^8inui ./ =0 dh -\- coaA^d^ -\- l^cos^ainA' .d^T+lb cos^sin^' ,f =zO dh + cos A", d^ + lbcosf sin A", d a T + 15 cos ^ sin A"./" = I (**®) &c. from which, by the method of least squares, the most probable values of dh, dip, and di^T are determined. The true values of the altitude, latitude, and clock correction will then be A + ^'A, <p + dip, ^T+d^T. The hour angles will be computed most accurately by (269)| which is the same as the following : tan« i t = ^^" K^ ^ ^ + ^) sin i (C + y — ^) cos i (C + f + ^) cos i (C — ^ — ^) in which ^ = 90° — h ; and the azimuths by tan« 1 A = »^" * ( ^ ~ y + ^) c os KC — y — ^) cos J (C + sp + ^) sin i (C + s^ — d) Since <p and ^ are constant, it wll be convenient to put then sin {c -\-\d) sin (ft — j ^) m = ; n = ^ cos (^ + i ^) cos (c — } ^) tan« i ^ = mn tan« M = - (851) The barometer and thermometer should be observed with each BY THREE OR MORE EQUAL ALTITUDES. 291 altitude, and if they indicate a sensible change in the refraction a correction for this change must be introduced into the equations of condition. Thus, if r^ is the refraction for the altitude h for the mean height of the barometer and thermometer during the whole series, while for one of the stars it is r, then the assumed altitude requires for that star not only the correction dA, but also the correction r — r^. Hence, if we find the refractions r, r', r", &c. for all the observations, and take their mean r^ we have only to add to the equations of condition respectively the quantities r — r^, r' — r^, r" — r^, &c. If any one of the stars is observed at an altitude \ slightly different from the common altitude A, we correct the correspond- ing equation of condition by adding the quantity h — \. 190. We may also apply the preceding method to the case where there are but three observations. The final equations are then nothing more than the three equations of condition them- Belves, from which the unknown quantities will be found by simple elimination. It will easily be seen that this elimination leads to the expressions iordf and d a T' already given on p. 284, if we there exchange dT^ dT'^ and dT^' for/,/', and/" respect- ively. We can simplify the computation by assuming a T so as to make one of the quantities /, /', /" zero. Thus, we shall have/= if we determine A^by the formula Ar=a + « — [T-j-arcr— ?;)] (352) then, finding/' and/" with this value, and putting ^ sin } A! cos 1 A! , "" sin 1 (.i' —A) sin } (A" t- A') ' ^ yf 9m\A'' QO%\A" ^ff "" sin } (A" — A) sin J (A" — A) we shall have the following formulae: di^T= — A:' sin J (A + A") -f A" sin \ (ii'+ A) -^^ = -^cosK^ + A'0 + ^'cos}(A'+A) ^ (353) .^-^ — = + jfc'cos J(A" — A) — A"cos J(ii'-- A) 15 cos f 292 LATITUDE. Example. — Taking the three observations above emplojed, and assuming the approximate values A T = — 11* ()•, ^ = 51^ 32' 0", A = 52<> 37' 0", we shall find, by (351), t = — 2» 36* 5-.50 t' == — 3» 19* 55*.65 ^ = 3* 23- 58*.25 A = — 66<> 16'.2 A' = — 177° 50'.2 A" = 90« 18M By (349), putting in this case 5 T = 0, we then have / = — 1-.83 /' = + 80-.95 /" = — d'.21 and the equations of condition (350) become dh + 0.4027 dip — 8.5410 <f a T + 15.63 = dh — 0.9993 dip — 0.3522 d a T — 28.51 = dh — 0.0053 d^ + 9.3308 d£iT— 57.94 = whence d^T= + 3-.92 dip = ^ 8".58 dh = + 21".81 and the true values of the required quantities are, therefore, A T = — 10* 56-.08 f = 51° 31' 51".42 h = 52° 37' 21".81 agreeing almost perfectly with the values before found. Since in this example there are but three observations, we may also employ the formulse (353), first assuming Ar= — 10-58M7 which is the value given by (352). With this we find /' = + 82'.78 /" = — 4v38 log ^ = 0.4199 log A" = nO.4932 and by (353) we shall find d^T= + 2'.09 cff = — 8".58 dh = + 21".31 Hence the true clock correction is — 10* 58M7 + 2'.09 = — 10* 56\08; and the values of the latitude and altitude also agree with the former values. BT TRANSITS. 293 191. We may here observe that, theoretically, the latitude might be found also from three different altitudes of the same star and the differences of azimuth ; for we should then have sin d = sin tp sin h -\- cos fp cos h cos A sin ^ = sin ^ sin K + cos ^ cos h' cos (A. -\- k) sin ^ = sin f» sin K' + cos ^ cos h" cos {A + k') in which A is the azimuth of the star at the first observation, and the differences of azimuth X and X' are supposed to be given. The solution of Art. 187 may be applied to these equations by writing h for d and A for t Again, there might be found from three different altitudes of the same star not only the latitude and time, but also the decli- nation of the star; for we then have sin k = sin ^ sin d -f cos ^ cos d cos t sin h' = sin ^ sin d -|- cos ^ cos d cos (t + '^ ) sin h" = sin f sin ^ -f cos ^ cos d cos (t + X') from which we can readily deduce ^, ^ and d. But the method is of no practical value, as the errors of observation have too much influence upon the result. NINTH METHOD. — BY THB TRANSITS OF STARS OVER VERTICAL CIRCLES. 192. We may observe the time of transit of a star over any vertical circle with a transit instrument (or with an altitude and azimuth instrument, or common theodolite) ; for when the rota- tion axis is horizontal, the collimation axis will, as the instru- ment revolves, describe the plane of a vertical circle. Tor any want of horizontality of the rotation axis, or other defects of adjustment, corrections must be applied to the observed time of transit over the instrument to reduce it to the time of transit over the assumed vertical circle. These corrections will be treated of in their proper places in Vol. 11. ; and I shall here assume that the observation has been corrected, and gives the clock time T of transit over some assumed vertical circle the azimuth of which is A. The clock correction a T being known, we have the star's hour angle by the formula 294 LATITUDE. and then, the declination of the star being given, we have the equation [from (14)] COS t sin f» — tan d cos f» = sin t cot A (354) If, then, A is also known, the latitude ip can be found by thia equation. Let us inquire under what conditions an accurate result is to be expected by this method. By differentiating the equation, we find [see (51)] cos fl' cos ^ .^ tan C , . . sin at ,^ dip = at a A -\ — ad cos C sin A sin A cos C sin A from which it appears that sin A and cos Q must be as great as possible. The most favorable case is, therefore, that in which the assumed vertical circle is the prime verHccd^ and the star a declination differs but little from the latitude ; for we then have A = 90° and (^ small. Indeed, these conditions not only increase the denominator of the coefficient of rf/, but also diminish its numerator, since, by (10), we have cos q cos ^ = sin C sin ^ 4~ ^^ ^ ^^^ 9 ^^^ ^ which vanishes wholly when the star passes through the zenith. Moreover, if the same star is observed at both its east and we:5t transits over the prime vertical, we shall have at one transit sin A = — 1, at the other sin J. == + 1, and the mean of the two resulting values of the latitude will, therefore, be wholly free from the effect of a constant error in the clock times, that is, of an error in the clock correction. It is then necessary only tliat the rate should be knowTi. This method, therefore, admits of a higli degree of precision, and requires for its successful applica- tion only a transit instrument, of moderate dimensions, and a time-piece. Its advantages were first clearly demonstrated by Bessel* in the year 1824 ; but it appears that verj' early in the last century Romer had mounted a transit instrument in the prime vertical for the purpose of determining the declinations of stars from their transits, the latitude being given. The details of this important method will be given in Vol. 11., under *' Transit Instrument.*' * Attronom, iVocA., Vol. IH. p. 9. BY TRANSITS. 295 193. It may sometimes be possible to observe transits only over some vertical circle the azimuth of which is undetermined. We must then observe either two stars, or the same star on opposite sides of the meridian. We shall then have the two equations cos t . tan A sin ^ — tan d . tan A cos f> = sin f cos f . tan ii sin ^ — tan d^ . tan A cos f = sin ^ from which the two unknown quantities A and f can be deter- mined. K the same star is observed, we shall only have to put d' = d. Regarding tan A sin f and tan A cos f as the unknown quantities, we have, by eliminating them in succession, - . sin t sin d' cos d — sin t' cos d' sin d tan ii sin ^ = cos t sin ^' cos d — cos f cos d' sin d ^ , — sin (f — t) cos d' cos d tan A COB^ = ^^ cos t sin d' cos d — cos f cos d' si sin d Jf we introduce the auidliaries m and Jf, such that m sin M= sin (d' + d) sin } (f — t) m cos M= sin (a' — d) cos i(f — t) } (355) we shall easily find m sin [} (^ + — -*n = 81^ * 81^ ^' c^8 9 — sin ^ cos d' sin 9 m cos [} (^ + — -^ = c^s * si"^ ^' cos d — cos f cos d' sin J m sin [} (f — t) — Jf] = — sin (t' — t) cos d^ sin ^ and hence tan il sin f = tan {_i'(f + t) — Jf] \ , , sin[J(^— — -^cot^ / (^^^) tan Acos0 = 5i_i^ ^ =* V cos [J («' + ^) — Jf] j which determine A and f> by a simple logarithmic computation. The solution will be still more convenient in the following form : ^ Bin («' — ») tan a» = tan d "-^ ^ ^ =L > r 357) 8in[i(<' — — J"] ^^ ^ _ tan[J(f + 0-^ sin f 296 LATITUDE. If the same star is observed at each of its transits over the same vertical circle, we have i' = ^, and hence tan Jif=QOy JIf == 90°, which gives tau f = tan d ^^ — ' — - tan A = ^^ — ■ — ^ (358) cos i (^ — ^^^ 9 K the same star is observed twice on the prime vertical, we must have <' + < = 0, since tan ^ = oo ; and then, . tan d tan d ,^.^. tan f = = (859) cos i (<' — t) cos t which follows also from (354) when cot ^ = ; or, geometrically, from the right triangle formed by the zenith, the pole, and the star, as in Art. 19. If the latitude is given, we can find the time from the transits of two stars over any (undetermined) vertical circle by the second equation of (357), which gives sin [}(«'+ — ^ = -^^^^ w° [i (^ — — -^] tan d for the observation furnishes the elapsed time, and hence V — i\ and this equation determines \{V + t\ and hence both i and f. If the latitude and time are given, \ye can find the declination of a star observed twice on the same vertical circle, by (358). When the observ^ation is made in the prime vertical, this becomes one of the most perfect methods of determining declinations. See Vol. n., TVansit Instrument in the Prime Vertical. 194. The following brief approximative methods of deter- mining the latitude may be found useful in certain cases. TENTH METHOD. — BY ALTITUDES NEAR THE MERIDIAN WHEN THE TIME IS NOT KNOWN. 195. (A.) By tico altitudes near the meridian and the chronometer times of the observations^ when the rate of the chronometer is known, but not its correction. Let h, K = the true altitudes, Tj T* ^^ the chronometer times, TWO ALTITUDES NEAR THE MERIDIAN. 297 then, t and t' being the (unknown) hour angles of the observations, we have, by (287), approximately, h^ = h' + af* in which h^ is the meridian altitude, and 225 sin 1" cos cos d a = ^ 2 cos \ The mean of these equations is and their diflference gives But we have T = i(!r'— T) = i(f—t) in which we suppose the interval T' — Tto be corrected for the rate of the chronometer. Hence 2 ar which, substituted in the above expression for h^, gives h, = i{h + h') + ar« + [^(^-"^')]' (360) According to this formula, the mean of the two altitudes is reduced to the meridian by adding two corrections: 1st, the quantity ar*, which is nothing more than the common "reduc- tion to the meridian*' computed with the half elapsed time as the hour angle ; 2d, the square of one-fourth the diflerence of the altitudes divided by the first correction. If we employ the form (285) for the reduction, we have in which h^ = i(h + A') + Am + ^^^^ ^ ^'^^' (361) Am . cos f cos d 2 sin' }t A ■== m = cos h^ sin 1" and m is taken from Table V. or log m from Table VI. 298 LATITUDE. Example 1. — ^From the observations in the example of Art 171, 1 select the following, which are very near the meridian. Obsd. alts. Q True alts. © Chronometer. 50° 5' 42".8 h' = 50^21' 7".6 23* 50- 4«'.5 50 7 25 .5 h = 50 22 50 .4 37.5 i(A-A')- 25.7 T= 4 55.5 Hh + h') = 50 21 59 .0 Am — + 59 .0 log m 1.6778 2d corr. = + 11 .2 log A 0.0930 A. = 50 23 9.2 log Am 1.7708 c, = 39 36 50 .8 log[j(A — V)]« 2.8198 i, - - 1 48 9 .2 log 2d corr. 1.0490 f = 37 48 41 .6 Example 2. — In the same example, the first and last observa- tions, which are quite remote from the meridian, are as follows: Obsd. alts. Q True alts. Q Chronometer. 49<> 51' 19".3 h = 50<> 6' 43".7 23» 37- 35- 49 50 24 h' = 50 5 48 .4 18 31 J(A — A')= 13 .8 T= 20 28 which give Am = 16' 58", and the 2d corr. = 0".2, whence ^ = 37° 48' 37". This simple approximative method may frequently be useful to the traveller, and especially at sea, where the meridian obser- vation has been lost in consequence of flying clouds. At sea, however, the computation need not be carried out so minutely as the above, and the method becomes even more simple. See Art. 204. M. V. Caillet* gives a method for the same purpose, which is readily deduced from the above. Put A =A' — A t'= r— T=2r then (360) becomes ' 2 4 4aT" 4ar" * TraM de Navigation (2d edition, Paris, 1S57), p. 819. THREE ALTITUSBS NEAR THE MERIDIAN. 299 or, patting sin 1" in which h is the altitude farthest from the meridian. Although this reduces the two corrections of (361) to a single one, the computation is not quite so simple. 196. (B.) By three altitudes near the meridian and the chronometer times of the observationSy when neither the correction nor the rate of the chronometer is knmon. — In this case we assume only that the chro- nometer goes uniformly during the time occupied by the observa- tions. Let A, A', h" = the true altitudes, T, T'f T" = the chronometer times, Tj =: the chronometer time of the greatest altitude. If we introduce the factor for rate = A, according to Art. 171, the formula for the reduction to the meridian by Gauss's method is, approximately, h^ = h + aki* in which t is the time reckoned from the greatest altitude. De- noting ak by a, we have then, from the three observations, \ = h'+aiT — T^y y (363) A^ = A"+a(T"-T,)« j which three equations suffice to determine the three unknown quantities a, T^, and hy By subtracting the second from the first, and the third from the second, we obtain T = a{T + T) — 2o7; A' A" - =a(r'-f T') — 2ar, rpn rp and the difference of these is A' — A" A — A' rptf rpf rpt rp = a{T"— T) 300 LATITUDE. If, then, we put b = — = the mean change of altitude in one second of the chronometer from the first to the second observation, A' — - A" ^ = ■;;;:; 7^. ^= ditto from the second to the third obser- fpff fpf vation, we have c — b rpn rp ^^ T+r b ^ ^ T+ T" c * 2 2a ' 2 2a (8&I) Having thus found T^, we can find h^ from any one of the equa- tions (363), all of which will give the same result if the compu- tation is correct.* Example. — Prom the observations in the example of Art 171 I select the following three observations : Obsd. alts. Q^ True alts. Chronometer. 50°5'42".8 h =60^21' 7".6 T = 23* 50- 46'.5 50 7 27 . A' = 50 22 51 .9 T' = 23 55 16 . 50 7 25 .5 h" = 50 22 50 .4 T" = 37 .5 A — A' = -- 104".3 T' — r = 269-.5 6 = — 0.8869 A' — A" = + 1 .5 T" —T= 321 .5 c = + 0.0047 T"— T=591. c — fe = + 0.3916 Kr+r') = 23.53. 1..3 log. =7^8 — — = + 4 52.0 log (T — T^y = 5.2604 T T^ ~ 23 57 53 .3 7; — 7- 6*.8 a(T- ioga(r r,y = 2.0817 A — 50° 21' 7".6 T,y — + 2 .7 A, — 50 23 8 .3 Cj — 39 36 51 .7 ^, — 1 48 9 .2 y = 37 48 42 .5 The moan of the throe values found from these altitudes in Art 172 is 37° 48' 42".8. * This method is estsentially the same as that proposed bj Littbow (Attrommnt, Vol. I. p. 171.) 1 hare here rendered it applicable to the sun without considering the change of declination, by introducing Gausses form for the redaction to the meridian. REDUCTION TO MERIDIAN BT AZIMUTHS. 801 197. (C.) By iioo altitudes or zenith distances near the meridian and the difference of the azimuths. — If the observer has no chrono- meter, he may still obtain his latitude by circummeridian alti- tudes, if he observes the altitudes with a universal instrument, and reads the horizontal circle at each observation, taking care, of course, that the star is always observed at the middle vertical thread. As this instrument generally gives directly the zenith distances, we shall substitute (^ for 90° — h. We have the equa- tion sin d = sin ^ cos C — cos f sin C cos A ^ sin (f — C) + 2 cos ^ sin C sin'Jil whence cos } (sp + ^ — C) sin } [C — (^ — ^)] = cos f sin C sin* } A But f — ^ = Ci = the meridian zenith distance; and hence . , ,_ -,. cos f sin Cain'} il ,^--^ sm } (C — Ci) = — (865) '^ ^^^ COS [^ - i(C - O] which expresses the reduction to the meridian = {^ — {^j when the absolute azimuth A is given. If the observation is very near the meridian, we may neglect J (^ — ^j) in the denominator of the second member, and take cos f sin Ci 2 sin* \ A * cos d sin 1" or, putting cos f sin Ci sin 1'' *i— ^ ■ ■ ■ ■ ■ ■■ ■ • w^^_^^^^Has « = " : ' • -^ (366) cos d 2 C — Ci = aA^ (367) from which it follows that near the meridian the zenith distance varies as the square of the azimuth. Now, when we have taken two observations, we have C, = C — aA^ Z,= t; —aA!' whence, putting 802 LATITUDE. we deduce tlie following equation, analogous to (360), Here r is equal to one-half the difference of the readings of the horizontal circle, and is therefore known ; and the computation is entirely similar to that of the formula (360). 198. (D.) By three altitudes or zenith distances near the meriditBH and the differences of azimuths. Supposing the observations taken with a universal instra- ment, let C, C, C = the true zenith distances, A, A', A" = the readings of the horizontal circle, we shall have, by the preceding article, C,= C -a(A -^)« ^ C,= r -a(A' ^A,y I (369) c, = r - a (A" - A,y j in which A^ is the (unknown) circle reading in the meridian, and a is the (unknown) change of zenith distance for 1" of azi- muth. These equations are solved in the same manner as (363); and hence we have the formulae = c = A' — A A" — A' c — h a = A'' — A A + A' h , A' + A" c A = — ! or A- = ' * 2 2a * 2 2a (870) which determine a and Ap after which {^jis found by any one of the equations (369).* * In this connection, see an article by Littsow in Zaoh*8 MotuUUcke CorrttpomiaUf Vol. X. (1824). BT CHANGE OF ALTITUDE. 803 ELEVENTH METHOD. — ^BY THE RATE OF CHANGE OF ALTITUDES NEAR THE PRIME VERTICAL.* 199. We have, Art 149, Ibdt = cos ^ sin A If then we observe two altitudes near the prime vertical in quick succession, noting the times by a stop-watch with as great pre- cision as possible, and denote the difference of the altitudes, or of the zenith distances, by rf{^, and the difference of the times by dty we shall have dz cos ip = cosec A (371) \bdt The observation being made near the prime vertical, an error in the supposed azimuth A will have but small influence upon the result. If the observation is exactly in the prime vertical, or within a few minutes of it, we may put 10 at This exceedingly simple method, though not susceptible of great precision, may be very useful to the navigator, as it is available when the sun is exactly east or west, and, consequently, when no other method is practicable, and, moreover, requires no previous knowledge of the time or the approximate latitude, or of the star's declination.f Example. — 1853 July 8, Prestel observed, near the prime vertical, the time required by the sun to change its altitude by a quantity equal to its apparent diameter, by observing with a sextant first the contact of the lower limb with its image in an artificial horizon, and then the contact of the upper limb with ♦ Pristil, in Attnm, Naeh., Vol. XXXVII. p. 281. f Since the star's declination is not required, this method has the additional adrantage (which maj at times be of great importance to the traTeller) of being practicable without the use of the EphemertM, This feature entitles this method to a prominent place in works on narigation. 804 LATITUDE AT SEA. its image, the sextant reading being the same at both observi- tions, namely, 30° 15' 0". He found Chronometer. Contact of lower limb, 4* 43* 34-. P.M. " upper " 4 47 5 .6 3 31.5 The sun's diameter was 31' 32". Hence we have d: = 31' 32" = 1892" log 3.2769 dt = 3" 31-.5 = 211-.5 ar. co. log 7.6747 log -fV 8.8239 ^ = 53^ 23'.5 log cos ip 9.7755 The azimuth, however, was not exactly 90°, but about 88® 20^. Hence we shall have, more exactly, 9.7755 A = 88<> 20' log cosec A 0.0002 y> = 53 22.3 log cos ^ 9.7757 It is evident that the method will be more precise in high lati- tudes than in low ones. FINDING THE LATITUDE AT SEA. First Method. — By Meridian Altitudes. 200. This is the most common, as well as the simplest and most reliable, of the methods used by the nax-igator. The alti- tude is observed with the sextant (or quadrant) from the sea horizon, and, in addition to the corrections used on shore, the dip of the horizon is to be applied. The true altitude being deduced, the latitude is found by (277) or (278), Art. 161. At sea the time is seldom so well known as to enable the navigator to take the star at the precise instant of its meridian passage. But the meridian altitude of a star is distinguished as the greatest, to secure which the observer commences to measure the star's altitude some minutes before the approximately com- puted time of passage, and continues to obsen^e it until he per- ceives it to be falling. The greatest of all his measures is then assumed as the meridian altitude. MERIDIAN ALTITUDES. 305 The most common practice in the case of the sun is to bring the lower limb, reflected in the mirrors of the instrument, to touch the sea horizon seen directly (a few minutes before noon), and then by the tangent screw to follow the sun as long as it rises, never reversing the motion of the screw ; as soon as the sun begins to fall, the limb will appear "to dip" in the sea by lapping over the line marking the horizon. Hence, when the sun "dips," the observation is complete, and the instrument is read oflT. But, as the waves of the sea cause the ship to rise and fall, the depression of the sea horizon is constantly fluctuating by the small amount due to the change in the height of the observer's eye : it is, consequently, impossible to keep the sun's reflected image in constant contact wdth the horizon. Expe- rienced observers advise, therefore, to observe and read oS separate altitudes in rapid succession, continuing until the numbers read off* decidedly decrease ; the greatest is then taken as the meridian altitude,* or, still more accurately, the mean of tlie greatest and the tT\'o immediately adjacent may be taken as the meridian altitude, free from the mequalities produced by the motion of the eye. 201. The greatest altitude, however, is not the meridian alti- tude, except in the case of a fixed star. To find the correction for a change of declination, we have, for the time (if) from noon when the sun is at the greatest altitude, the formula (294), or tf = A^ sin (<p — d) 810000 sin 1" cosy^cos^ in which ^d is the hourly change of declination expressed in seconds. The reduction of the maximum altitude to the meri- dian altitude is the quantity y. Art. 172, or (15 dy sin 1" cos ^ cos^ 2 sin (f — d) These formulse give t? in seconds of time and y in seconds of arc. For nautical use, let a = the change of altitude (expressed in seconds of arc) in one minute of time from the meridian ; * Rapib, Practice of Navigation (4th edition, 1852), p. 226. Vol. L— 20 306 LATITUDE AT SEA. then, by (287), putting / = 60-, 810000 sin 1" cos ^ cos ^ a = 2 sin (f — d) and therefore (873) 2a ""Uo/ 4a The value of a is given in Bowditch's Navigator, Table XXXIL, with the arguments <p and d. If, xoe express a5 in minutes of arc^ we shall have t? in minutes of time and y in seconds of arc, by the formulae* * = — y = - — — r374) These formulse may be used also for the moon or a planet. The greatest value of a5 for the sun is 1', namely, at the equinoxes when 5 = 0; and in this case, if the latitude is 70°, wo have a = 0.7 and 1" y = — == 0".36 ^ 4 X 0.7 a quantity altogether insensible in nautical practice. For the moon, however, we may have ^8 = 18', and for f = 70° the least value of a = 0.6, whence = ^'^^y = 135" = 2' 16" ^ 4 X 0.6 Even this (which, it must be observed, is for an extreme case) is usually neglected by navigators, who regard observations of tlie moon for latitude as but approximations, on account of the frequent indeterminate character of the sea horizon as seen under the moon.f 202. When the ship is in motion, the change of latitude pro- duces the same effect upon the obscrv^ed maximum altitude as an equal change of declination. Thus, as in the last example of the preceding article, if a ship in latitude 70° sail due north ♦ BowDiTCH, Practical Navigator^ p. 169. t Rapkb, Practice of Navigation (4th edition), pp. 177, 226, 230. REDUCTION TO THE MERIDIAN. 807 or due south at the rate of 18 miles per hour, the maximum altitude will exceed the meridian altitude by 2' 15". Second Method. — By Reduction to the Meridian when the Time is given. 203. "When the meridian observation is lost in consequence of clouds, circummeridian altitudes may sometimes be obtained. The most convenient method of reducing them at sea is that of BowDiTCH. In his Table XXXTL he gives the value of a com- puted by (373); and in Table XXXIII. the value of t\ t being' reduced to minutes. Each observed altitude h is then reduced to the meridian altitude hy by the formula (287), or h^ = h + ai^ (376) and a number of altitudes are reduced at once by the same formula, by taking for h the mean of all the altitudes, and for t^ the mean of all the values of t^. If the observer has no tables, he can readily compute a by the formula a = l".9635 r,t^* = [0.2930] £S!!^^ (376) 8m(f> — d) "" ■'8m(f> — d) ^ ^ Bowditch's table for f^ extends, however, only to t = 13"*. When the observations are more than 13"* from the meridian, he reduces the observation to the meridian by the formula (282), cos Ci = sin A 4" cos f> cos d (2 sin* } f) employing a table of log. versed sines for the value of 2 sin* J^; a table of natural sines for sin h and cos f ^ ; and the table of logarithms of numbers for the value of the last term. I prefer the formula (283), . , ,, ,v cos cp cos ^ sin* } < sm 1 (A, — A) = ^ ^ cos J (A, + A) which effects the reduction by a single table. Third Method. — By Two Altitudes near the Meridian when the Time is not known. 204. As it frequently happens at sea that the local time is uncertain, the method I have proposed in Art. 195 will be found ■I ■ ^1* I 308 LATITUDE AT SEA. of great use to the navigator. Any two altitudes h and h' being observed near the meridian, r being one-half the chronometer interval between them, corrected for rate, expressed in minutes, and a being found by (376), or from Bowditch*s Table XXXIL, we have the meridian altitude by the formula which may be computed without the use of logarithms. Example. — The approximate latitude being 38® N., the de- clination at noon 1° 48' 9" S., the height of the eye above the sea 19 feet, suppose the following observations taken : Chronometer. Q 3" — 8» 0-22'.5 h' — 50« >10' 0" T — 8 10 13 .5 h =50 11 40 2) 9 51 h — h' = 1 40 T — 4 55.5 l(A-A')- 25 T« — 24.2 }(A + A') — 50 10 50 a = 2".4 AT* — Ist corr. — 58 \\(h. - h')y = 625 \V 2d « 11 Mend. alt. — 50 11 59 Dip — 4 16 Semidiamctcr — + 16 6 Eefr. and par. = — 42 A, — 50 23 7 C, -39 36 53 N. i, 1 48 9 S. SP = 37 48 44 N. Tlie accuracy of the result depends in a great degree upon the accuracy with which the difference of altitude is obtained. If in t!ie above example this difterence had been 2' 40", or 1' too great, we should have found J(A — A') = 40", and the 2d correction — ajJ*^- -- 28" : consequently the resulting latitude would have been only 17" too small. Since the same causes of error, such as displacement of the sea horizon by extraonlinary refraction, unknown instrumental errors, &c., aftect both altitudes alike, the difference will usually be obtained, even at sea, within a quantity much less than V. The most favorable case is that THREE ALTITUDES NEAR THE MERIDIAN. 809 in which the altitudes are equal and the 2d correction, conse- quently, zero. It will be well, therefore, always to endeavor to obtain altitudes on opposite sides of the meridian. We may also obtain an approximate value of the time from the same observation ; for we have for the hour angle of the least altitude h\ Art 195, a = ^^^-^^ + 1(2"- T) or Thus, in the above example we have i(A-AO ^ 25 ^ 2«1 ar 2.4 X 4.9 1(7'— r) = + 4 .9 f = + 7 .0 The apparent time of the observation of the least altitude was, therefore, 0* 7*. Fourth, Method. — By Three. Altitudes near the Meridian when the Time is not known. 205. The method of Art. 196 does not require even the rate of the chronometer to be known ; but it is hardly simple enough for a common nautical method. But a very simple method will be obtained if we take three altitudes at equal intervals of time. Suppose the second altitude is observed at the (unknown) time Tivora the meridian passage, the first at the time T— Xy the third at the time T + x; then we have, by (363), \=h -j-a(T—xy \=h'' + a(T+xy Subtracting the half sum of the first and third equations from the second, we deduce 810 LATITUDE AT SEA. The difference of the first and third gives X which substituted in the second equation gives Aj. If then we put a for cu?^ the computation is expressed by the following simple formulae : a =A'— 1(A + A") (877) ^ ' a Example. — The following three altitudes were observed at equal intervals of time near the meridian : h = 43^ 8' 20" h! = 43^ 15' 30" h" = 43^ 4' 0" i(A + A")=43 6 10 a = 9 20 = 660" \(h — K')= 1 6= 65 Hence the reduction of the middle altitude to the meridian is [\(h^r)Y ^ 65« _ g, a 560 which added to h' gives h^ = 43*' 15' 38" Instead of equal inter\'al8 of time, we may employ equal inter- vals of azimuth (Art. 197), and still reduce the altitudes by (377); but this would be practicable only on land. Fifth Method, — By a Single Altitude at a given Time. 206. This is the method of Art. 164, which, however, should be restricted, at sea, to altitudes taken not more than one hour from the meridian, as the time is always imperfectly known and ALTITUDE NEAR THE PRIME VERTICAL. 811 the error in the latitude produced by an error in the time increases very rapidly as the star leaves the meridian and ap- proaches the prime vertical (Art. 166), and the method fails altogether when the star is in the prime vertical. It may, how- ever, sometimes be verj' important to determine the latitude, at least approximately, when the sun is nearly east or west; and then the follo>ving method may be used. Sixth Method. — By the change of Altitude near the Prime Vertical 207. This is the method of Art. 199. In the morning, when the sun has arrived within 1° of the prime vertical as observed with the ship's compass, bring the image of the sun's upper limb, reflected by the sextant mirrors, into contact with the sea horizon, and note the time ; let the sextant reading remain un- changed, and note the time when the contact of the lower limb occurs. In the afternoon, begin with the lower limb. Then, taking the sun's semidiameter = S from the almanac, and put- ting the difierence of the chronometer times = r, we have cos sp = — = [9.1249] - (378) 15 r ^ This is evidently but a rough method, only to be resorted to in cases of emergency. With the greatest care in observing the contacts, and in latitudes not less than 45°, the result cannot be depended upon within from five to ten minutes; but even this degree of accuracy may, in many cases at sea, be quite satis- fiBictory. Seventh Method, — By the Pole Star. 208. This method, though confined in its application to north latitudes, is very useful at sea, as it is available at all times when the star is visible and the horizon sufliciently distinct, and does not require a more accurate knowledge of the time than is usually possessed on shipboard. The complete discussion of it has been given in Art. 176 ; but for those who wish only the nautical method, and have passed over that article, I add the following simple investigation, which is sufliciently precise for the purpose. Let ZNy Fig. 27, be the meridian ; Z the zenith of the ob- server ; P the pole ; AN the horizon ; S the star, which describes 312 LATITUDE AT SEA. a small circle ST about the pole at the dis- tance PS=p; ZSA the vertical circle of the star at the time of the observation; SA the true altitude = A, deduced from the observed ; SPZ the star's hour angle = t ; PN the lati- tude = (p. Draw SB perpendicular to the meridian: then, since SP is small in the case of the pole star (about 1° 30'), we may regard PSB as a plane triangle, and hence we have PB = PS, COB SPB =pcost and, since BN differs very little from SA, that is,* K we put we have and hence PN= ^i\r— PB = SA — PB <p =zh — p cos t = the sidereal time, a = the star's right ascension, t= e —a fp = h — ;) cos (6 — o) (379) If then p and a be regarded as constant, the term p cos (0 — a) may be given in a table with the argument 0, as in Bowditch's Navigator, p. 206. But the polar distance and right ascension of the pole star vary so rapidly that in a few years sucli a table affords but a rude approximation. The direct computation of the formula with the values of p and a obtained from the Ephemcris for the day of the observation is preferable. Example. — 1856 March 10, from an altitude of Polaris ol)- served from the sea horizon, the true altitude h was deduced as below. The time was noted by a Greenwich chronometer which was fast 5"* 30*. The longitude was 150° 0' W. * If we compare this with the more exact formula (300), we see that the error of the nautical method is J p^ sin 1" sin^ t tan h, which is a maximum for t = 90^. Taking p .= 1® 30', this maximum is 70''.7 tan ^, which amounts to 3' when p -^ 68® 30'. BY TWO ALTITUDES. 813 Chronometer 19» 12- 42* h = SV W. Correction — 6 30 Gr. M. T. 19 7 12 p = V 27' 18" Longitude * 10 = 8r.3 Local M.T. 9 7 12 logjp 1.9410 Sid. T. Gr. noon 23 13 23 log cos t n9.5234 Corr. for 19^ 7* + 3 8 log;>co8f nl.4645 — j)C08f= + 29.1 e= 8 23 43 f = 31 39.1 a= 1 5 44 t= 7 17 69 = 109<'2y 45" Eighth Method. — By Two Altitudes with the elapsed Time between them, 209. This method may be successfully applied at sea, and" is the most reliable of all methods, next to that of meridian or cir- cummeridian altitudes. The formulre fully discussed in Arts. 178 to 183 may be directly applied when the position of the ship has not changed between the obsenrtitions. But, since there should be a considerable difference of azimuth between the observations, the change of the ship's position in the interval will generally be sufficiently great to require notice. All that is necessary is to apply a correction to the altitude ob- served at the first position of the ship, to reduce it to what it would have been if observed at the second position at the same instant To obtain this correction, let J^', Fig. 28, be the zenith of the observer at the first observa- tion, S the star at that time ; Z his zenith at the second observation, and S' the star at that time. The first observation gives the zenith distance Z'S^ the second the zenith distance ZS\ Joining the points S and S' with the pole P, it is evident that the hour angle SPS' is obtained from the observed difference of the times of observation precisely as if the observer had been at rest. We have, there- fore, only to find ZS in order to have all the data necessary for computing the latitude of Z by the general methods. The number of nautical miles run by the ship is the number of minutes in the arc ZZ' ; and, since this will always be a suffi- 314 LATITUDE AT SEA. ciently small number, if we draw ZA perpendicular to SZ\ we may regard ZAZ' as a plane triangle, and take ZS = Z'S — AZ' or ZS = Z'S — ZZ' COS ZZ'S (380) The angle ZZ^S is the diftercnce between the azimuth of the star at the first observation and the course of the ship; and this azimuth is obtained with sufficient accuracy by the compass.* Employing the zenith distance thus reduced and the other data as observed, the latitude computed by the general method will be that of the second place of observation. In the same manner we can reduce the second zenith distance to the place of the first, and then the latitude of the first place will be found. 210. The problem of finding the latitude from two altitudes is most frequently applied at sea in the case where the sun is the observed body, the observation of the meridian altitude having been lost. The computation is then best carried out by the fomml^e (315), (316), (317), (318), employing for 3 the mean declination of the sun, — i.e. the declination at the middle time between the two obser\^ations, — and then applying to the result- ing latitude the correction a^ found by the formula (323). To save the navigator all consideration of the algebraic signs in computing this correction, it will be sufficient to observe the following rule : 1st. When the second altitude is the greater^ ^pply this correction to the computed latitude as a northing when the sun is moving towards the norths and as a southing when the sun is moving towards the south ; 2d. When the first altitude is the greater, apply the correction as a southing when the sun is moving towards the north, and as a northing when the sun is moving towards the south, * If wc wish a more rigorous process, we must consider the sphericftl triangle ZZ'Sj in which we have the observed zenith distance Z'S .^ (C). the required lenith distance ZS -.= C, the distance run by the ship Z'Z = </, the di£fcreDce of the star's azimuth and the ship's course ZZ'S' = a, and hence cos ^ = COS ^' COS d -\- sin ^' sin d cos a which deTeloped gives (i = C — d cos a -f J ^ sin 1" cot ^' sin* a the last term of which expresses the error of the formula given in the text. BY TWO ALTITUDES. 815 If the computer chooses to neglect this correction, he should employ the mean declination only when the middle time is nearer to noon than the time of the greater altitude. In all other cases he should employ the declination for the time of the greater altitude (Art. 183). 211. DouwEs's method of ^^ double altitudes.''* — This is a brief method of computing the latitude from two altitudes of the sun, which, though not always accurate, is yet sufficiently so when the interval between the observations is not more than 1*, and one of them is less than 1* from the meridian. Let A and h' be the true altitudes, d the declination at the middle time, T tmd T' the chronometer times of the observa- tions, i and V the hour angles. The elapsed apparent time X is found from the times 7 and T' by (322), but it is usually suffi- cient to take X=^T—T. We then have V=t^rX\ and by the first of (14) we have sin A = sin ^ sin d -f- cos ip cos 5 cos t sin K = sin f> sin ^ + cos ^ cos d cos (t -f X) The difference of these equations gives sin h — sin A' = 2 cos ^ cos d &\n (t -\- i X) sin iX If we put i^ = the middle time, or t,= t-\-hX we deduce „ . . sin A — sin A' .^o-. 2 sm fj, = (381) cos ^ cos ^ sin i A which gives t^ by employing the supposed latitude for ip in the second member. We then have t = % — \X and the meridian zenith distance fj is found from the greater altitude A by the formula (Art. 168) ' cos Ci = sin A 4- cos <p cos d (2 sin* 1 1) * The method of finding the latitude by two altitudes is commonly called by nayi- gators '*the method of double altitudes/' — an obvious misnomer, as double means twice the same. 316 LATITUDE AT SEA. and finally the latitude by the formula y = Ci + ^- Since we employ an assumed approximate latitude, we shall have to repeat the process when the computed latitude differs much from the assumed. This is the form of the method as proposed by Douwes and adopted in Bowditch's Navigator; but the following form is still more simple, as it requires only the table of logarithmic sines. The formula for t^ may be written thus : cos i (A + h') siniCh — h') sm t^ = ^^ — ' ^ ^^ ^ cos ^ cos ^ siniil then, as before, and the reduction of h to the meridian altitude h^ is found by (283), . , ,, , V cos <p cos ^ sin* it sm 1(\ -h)= cos i (A, + A) Adding h^— hto A, we have the meridian altitude, from which the latitude is deduced in the usual manner. If the greater altitude is within the limits of circummeridian altitudes, it will of course be reduced by (284). The chief objection to this method is that the computation must be repeated when the assumed latitude is much in error. It can also be shown that unless the observations are taken as near to the meridian as we have above supposed, the computed value of the latitude may in certain peculiar cases be more in error than the assumed value, so that successively computed values will more and more diverge from the truth. The methods referred to in the preceding articles are, therefore, generally to be preferred. 212. The latitude may also be found from two altitudes by the simple method proposed by Captain Sumner, for which see Chapter Vm. BY CHBONOMETEBS. 817 CHAPTER Vn. FINDING THE LONGITUDE BY ASTRONOMICAL OBSERVATIONS. 213. The longitude of a point on the earth's surface is the angle at the pole included between the meridian of the point and some assumed ^r^^ meridian. The difterence of longitude of any two points is the angle included by their meridians. These definitions have been tacitly assumed in Art. 45, where we have established the general equation £ = 2; — T (382) in which (Art. 47) 7J, and T are the local times (both solar or both sidereal) reckoned respectively at the first meridian, and at that of any point of the earth's surface, and L is the west longitude of the point. As an astronomical question, the determination either of an absolute longitude from the first meridian, or of a difterence of longitude in general, resolves itself into the determination of the difference of the time reckoned at the two meridians at the same absolute instant.* The various methods of finding the longitude which are treated of in this chapter differ only in the mode by which the comparison of the times at the two meridians is efiected. FIRST METHOD. — BY PORTABLE CHRONOMETERS. 214. The difference of longitude between two places A and B being required, let a chronometer be accurately regulated at A^ that is, let its correction on the time at that place and its daily rate be determined by the methods of Chapter V. ; then let the chronometer be transported to -B, and let its correction * The astronomical difference of longitude may differ Arom the geodetic difference for the same reason that the astronomical latitude differs from the geodetic, Arts. S6 and 160. 818 LONGITUDE. on the time at that place be determined at any instant. The time reckoned at A at this last instant is also known from the correction and rate first found, provided the rate has not changed in transportation; and hence the difference of times at the same absolute instant, and consequently the difference of longitude, are found. Let aT, dT=the correction and rate determined at A at the time T, by the chronometer, aT' = the correction determined at B at the time T' = T -\- tj t being the interval by the chro- nometer ; then, at the instant T+ t the true time at^is T +t + liT+t.dT '' B T -\-t + aT' and hence the difference of longitude is L = £^T -^-t.dT—i^T (383) Thus, the longitude is expressed as the difference of the two chronometer corrections at the two places; and the absolute indications of the chronometer do not enter, except so far as they may be required in determining the inter\'al with which the accumulated rate is computed. In this expression dT\^ the rate in a xniit of the chronometer (an hour, or a day, solar or sidereal), and T' — T must be expressed in that unit. Example. — At Greenwich, May 5, mean noon, a mean time chronometer marks 23* 49"* 42*.75, and its rate in 24 chronometer hours has been found to be gaining 2'.671. At Cambridge, Mass., May 17, mean noon, the same chronometer marks 4* 34* 47'.28; what is the longitude of Cambridge ? We have T= May 4, 23* 49" 42v75 a T= + 0* 10- 17-.25 ^7= — 2*.671 T+f= " 17 , 4 34 47.28 t= 12" 4»45- 4*. 53 = 12^.198 Ilence A!r+f.^!r= + 0* 9-44«.67 Ar = ~4 34 47.2 8 i=:_^4 44 31.95 BY CHRONOMETERS. 819 KoTS. — It is proper to distinguish whether the given rate is the rate in a chrono- meter unit or in a true unit of time; although the difference will not be appreciable unless the rate is unusually great. If the rate is 20* in 24* by the chronometer, it will be 20" =i: COOd in 24* of solar time. 215. When the chronometer is carried from point to point without stopping to rate it at each, it is convenient to prepare a table of its correction for noon of each day at the first station, from which the correction for the time of any observ^ation at a transient station may be found by simple interpolation. After reaching the last station, it is proper to re-determine the rate, which will seldom agree precisely with that found at the first. In the absence of any other data aftecting the rate, we may assume that it has changed uniformly during the whole time. It is convenient to compute the longitudes first upon the supposition of a constant rate, and then to correct them for the variation of rate, as follows. Let AT,dT= the correction and rate at the time T, found at the first station, d'T=the rate found at the last station at the time and put d'T—dT X = — — (384) n then X is the increase of rate in a unit of time. If an observa- tion at an intermediate station is taken at the time 7^ + ^ we must compute the accumulated rate for the interval /, which is eftected by multipljdng the mean rate during this interval by the interval. But, upon the supposition of a uniform increase, the mean rate from the time T to the time T + iis the rate at the middle instant T -\- ^t, and this rate is dT+ \tx. Hence the chronometer correction on the time at the first station at the instant T -\- ioi the supposed observation is £^T-\-t{dT+ \tx^ = £^T-\-t.dT+ \Vx (385) A longitude assigned to an intermediate station at the time T + i^ by employing the original rate ST^ will therefore require the correction + ^t^x, observing always the algebraic signs of x and the longitude. 320 LONGITUDE. If a number m of chronometers have been employed, and each determination of a longitude is the mean of the 97) values which thej have severally given, the longitude assigned upon the sup- position of constant rates is to be corrected by the quantity V X, + X, + X, + &c. + x^ 2 m in which x^ x^ &c. are the increments of the rates of the several chronometers in a unit of time. If then we put s = the sum of all the total increments during the whole interval n, or the sum of the values oi d'T — ^Tfor the several chronometers, 8 q = 2mn •we shall have Correction of a longitude at a time T + t = f.q (386) Example.* — ^In a voyage between La Guayra and Carthagena, calling on the way at Porto Cabello and Cura9oa, the following observations having been made, the relative longitudes are re- quired. By observations at La Gua}Ta on May 22 and 28, the cor- rections and rates of chronometers F, 3/, and P at the mean epoch May 24*'.885 were as follows : A^ ST Chron. F. — 4*33- 7'.80 + 0-.77 M, — 4 17.40 — 4.54 P. — 69 43.70 — 1.47 On arrival at Porto Cabello, the corrections on the mean time at that place on June &'.870 were ascertained to be — L'T F. — 4» 37- 15'.80 M. 4 5 31.28 P. — 5 14 13.38 At Cura^oa the corrections on June 12^.890 were — * Shadwill, Notes on the Management of Chronometert, p. 111. BT CHRONOMETERS. 821 a'T F. — 4» 40- 59'.20 M. — 4 9 55.53 P. — 5 18 8.24 And finally, at Carthagena, observations on the 25tli and 29th of June gave the corrections and rates at the mean epoch June 27*'.0 as follows : A'T i'T F. — 5» 7-23'.55 + 0'.85 M. 4 37 47.98 — 5.90 P. — 5 44 34.42 + 0.30 Employing the rates found at La Guayra, the corrections of the chronometers on June 5**. 870 at Porto Cabello (for which we have t = ll^'.OSS), and the resulting difterence of longitude, are, by formula (383), are as follows : ^T+t.iT P. Cabello — L» QoayTa. F. — 4» 32- 58'.57 + 4- 17'.28 Jf. — 41 11.81 19.47 P. — 5 10 1.82 12.06 Mean + 4 16.25 With the same rates, we have on June 12.890 at Cura^oa (for t == 19**. 005) the corrections and the corresponding difterence of longitude, as follows: /iT+t.iT Ciink9oa — Lft Ooayra. F. — 4» 32- 53M7 + 8- 6'.03 M. — 4 1 43.68 8 11.86 P. — 4 10 11.64 7 51.60 Mean + 8 3 .16 With the same rates, we have on June 27** at Carthagena (for i = 8S*.115) the corrections and the corresponding difterence of longitude, as follows: F. dkT+ t.iT Cutbagena— La OaayTa. — 4» 32- 42'.30 + 34- 41'.25 M. 4 2 47.74 35 0.24 P. — 5 10 32.38 34 2.04 L— 21 Mean + 84 34 .51 322 LONGITUDE. Now, to correct these results for the changes in the rates of the chronometers, we have, in the interval n = 83.115, 6'T—6T F. + 0*.08 M. — 1 .36 P. +1 .77 and, consequently. 5 = + .49 ^ -1-^'^^ ^ ^ 0'.002466 ^ 2 X 3 X 33.115 ^ Applying the correction t\ to the several results, the true differences of longitude from Ea Guayra are found as follows: Approx. diff. long. t^.q Corrected diff. long. P. Cabello + 4- 16-.25 + 0*.35 + 4- 16'.60 Cura^oa +8 3 .16 + .89 +8 4 .05 Carthagena + 34 34 .51 + 2 .70 + 34 37 .21 But it is usually preferable to carry out the result by each chronometer separately, in order to judge of the weight to be attached to the final mean by the agreement of the several indi- vidual values. For this purpose we have here, by the formula (384), for n = 33.115, F, + 0.00121 M, — 0.02054 P. + 0.02673 and hence the correction ^t^.x is, for the several cases, as follows : p. Cabello. Cura9oa. Carthagena. F. + 0M7 + 0-.44 + 1-.32 M, —2.95 —7.41 —22.52 P. + 3 .84 +9 .65 + 29 .31 Applying these corrections severally to the above approximate results, we have, for the differences of longitude from La Guayra, V. Cabello. Curafoa. Carthagena. F. + 4- 17'.40 + 8- 6'.47 + 34- 42'.57 M. 16.52 4.44 37.72 P. 15.90 1.25 31.35 Means + 4 16 .61 +84 .05 + 34 37 .21 agreeing precisely with the corrected means found above. BY CHRONOMETERS. ^23 K the chronometers have been exposed to considerable changes of temperature, the proper correction may be intro- duced by the method of Art. 223. 216. Chronometric expeditions between itoo points. — ^Where a dif- ference of longitude is to be determined with the greatest possible precision, a large number of chronometers are trans- ported back and forth between the extreme points. There are two classes of errors of chronometers which are to be eliminated: Ist, the accidental errors, or variations of rate which follow no . law, and may be either positive or negative; 2d, the constant errors, or variations of rate which, for any given chronometer, appear with the same sign and of the same amount when the chronometer is transported from place to place ; in other words, a constant acceleration, or a constant retardation, as compared with the rates found when the chronometer is at rest. The accidental erroi's are eliminated in a great degree by employing a large number of chronometers, the probability being that such errors will have different signs for different chronometers. The constant errors cannot be determined by comparing the rates at the two extreme points, since these rates are found only when the chronometer is at rest ; but if the chronometers are trans- ported in both directions, from east to west and from west to east, a constant error in their travelling rates will affect the differ- ence of longitude with opposite signs in the two journeys, and will disappear when the mean is taken. These considerations have given rise to extensive expeditions, of which probably the most thoroughly executed was that carried out by Struve, in 1843, between Pulkova and Altona.* In this expedition sixty- eight chronometers were transported eight times from Pulkova to Altona and back, making sixteen voyages in all, giving the difference of longitude between the centre of the Pulkova Obser- vatory and the Altona Observatory^ 1* 21'* 32*. 527, with a probable error of only 0'.039. Chronometric expeditions between Liverpool (England) and ♦ Expedition ehronomitrique exSeutSe par ordrt dt Sa Majesti UEmpereur Nicolas I. pour la dilermination de la longitude giographique relative de V ohtervatoire central de Ru99ie. St. Petersburg, 1844. For an account of the carefully executed expedition under Professor Aibt to deter- mine the longitude of Valentia in Ireland, see the Appendix to the Greenwich Observations of 1845. LONQIirDE. Camhriilgti (U. 8.) were instituted in the years 1849, '50, '51, ami "55 by the U. 9. Coast Survoy, niider the Bupcrintctnlciico of Professor A. D. Bache, The results of the expeditions of 1849, '50, and '51, diseusaed by Mr. G. P. BosD,* proved the neeefisity of introducing a correction for the temperature to which the chronometers were exposed during the voyages, and partic-oW attention was therefore paid to this jwint in the expetlitioii of 1855, the details of which were arranged by Mr. W. C. Boxo. The results of six voyages, — three in each direction, — according to the clisciiseion of Mr. Q. P. Bosd,-|- were as follows : Voyages from Liverpool to Cnmbridgo 4' 32" 31'.92 " " Cambridge to Liverpool 4 32 81 .75 Mean 4 S2 31 .&4 with a probable error of 0'.19. In this expedition fif^ chrono- meters were used. The greater probable error of the result, u compared with Strdve's, is sufficiently explained by the greater length of the voyages and their smaller number. 217. The following is essentially Steuvk's method of conduct- ing the expeditions and discussing the results. Before embarking the chronometers at tlie first station {A), they arc carefully compared with & standard clock the correction of which on the time at that station has been obtained with the greatest precision by transits of well-determined stare. (See Vol, n., " Transit Instrument.") Upon tlieir arrival at the aecond station (^, they are compared with the standard clock at that station.! From these two comparisons the chronometer correc- tions at the two stations become known, and, if ihe rates ant known, a value of the longitude is found by each chrononivtor by (383). But here it is to be observed tliat the rate of a chro- nometer is rarely the same when in motion as when at re«L It is necessary, therefore, to find its travcUinff rate (or wa raU, as it is called when the chronometer is transported by sea). Thi* might be efl'ecfed by finding— ^rst, the correction of the chrono- * Report of ibc Superintendent of Ihe U. S. CoibI Siirre; for I8A4, Appendix N*. 43. f Repari of Lbe SupKrintendcni of Uie U. S. Cowl Survcj for 1^00, p. 182. t Fur thp method of comparing ehronomclen ftnd clucki with tlie x'VWcM p««- cUion, ate Vol. U, BY CHRONOMETERS. 325 meter at the station A immediately before starting ; secondly^ its correction at B immediately upon its arrival there ; and thirdly, having, without any delay at By returned directly to A, finding again its correction there immediately upon arriving. The dif- ference between the two corrections at A is the whole travelling rate during the elapsed time, and this rate would be used in making the comparison with the correction obtained at -B, and in deducing the longitude by (383). But, since the chronometer cannot generally be immediately returned from -B, its correction for that station should be found both upon its arrival there and again just before leaving, and the travelling rate inferred only from the time the instrument is in motion. For this purpose, let us suppose that we have found at the times f, if, <", f", the chron. corrections a, 6, ft', a\ the correction a at the station A before leaving ; b upon arriving at B; b' before leaving B; and a' upon the return to A. The times /, t\ V'y V"j being all reckoned at the same meridian, if we now put m = the moan travelling rate of the chronometer in a unit of time, k = the loDgitude of B west of A, we shall have, upon the supposition that the mean travellini^ rate is the same for both the east and west voyages, X = a +m(f ^t) — b >l = a'— m(r— r) — 6' From these two equations the two unknown quantities m and k become known. Putting r=zf—t T"=r— r we find, first, m = (^'-^)-(y-^) (387) in which the numerator evidently expresses the whole travelling rate, and the denominator the whole travelling time. Then, patting 826 LONQITUDE. (a) = a + ^^ we have } (388) A = (a) — 6 in which (a) is the interpolated value of the chronometer correc- tion on the time at JL, for the same absolute instant (^ to which the correction 6 on the time at B corresponds. Example. — ^In the first two voyages of Struve's expedition between Pulkova and Altona in 1843, the corrections of the chronometer "Hauth 31" were found, by comparison with the standard clocks at the two stations, as below. The dates are all in Pulkova time, as shown by one of the chronometers em- ployed in the comparison : At Pulkova (A), t = May 19, 21*.54 a = + 0» 6- 38M0 "Altona lB),f = « 24,22.66 b=--l 14 39.92 " Altona (J?), r = « 26, 10 .72 6' = — 1 14 36 .77 " Pulkova (il), r = « 81, 0.00 a'= + 7 9.M ITence T = 5' 1M2 = 5'.047, a''^a = + 81-.48 t"=4 13.28 = 4.663, 6' — 6 = + 8.15 8P.48 — 3M5 28*.33 . ^ ^^, m = = = + 2-.961 6.047 + 4.563 9.6 a = + 0* 6* 38-.10 mrz= 4- 14.89 (a) = + 6 52.99 b = ^l 14 89.92 X = (a)—h = + l 21 82 .91 218. In the above, the rate of the chronometer is assumed to be constant, and the problem is treated as one of simple inter- polation. But most chronometers exhibit more or less accelera- tion or retardation in successive voyages, and a strict interpola- tion requires that we should have regard to second differences. If we always start from the station Ay as in the above example, using only simple interpolation, we commit a small error, which always affects the longitude in the same way so long as the variation of the chronometer's rate preserves the same sign. But if we commence the next computation with the station -B, BY CHRONOMETERS. 827 SO that the two chronometer corrections' at A are intermediate between the two at -B, then the error in the longitude will have a different sign, and the mean of the two values of the longitude will be, partially at least, freed from the influence of the acce- leration or retardation. To show this more clearly under an algebraic form, let us suppose that we have, omitting the inter* vals of rest at the two stations. at the times t, f, <", f", the chron. corrections a, b, «', V, intervals ^, t", and that (I = daily rate of the chronometer at the time t, 2fi = the daily acceleration of the rate fi after the time t, the true values of the four corrections, observing that b and 6' refer to the meridian of By will be, according to the law of uni- formly accelerating motion, a = a b ^a + fAT + fit'—X If now we find the value of (a) corresponding to b (that is, for the time (') by simple interpolation between the values of a and a% we have « = «+(7^) = a + Air + /9.T(T + 0' from which we obtain the erroneous longitude X'=(^d)—b = X+fiTr^ Hence the error in the longitude, by simple interpolation and commencing with the station JL, is dX' = ^rr'. In the next place, if we commence at the station B^ with the correction 6, employing simple interpolation between 6 and 6', to find the correction (6) for the time /" corresponding to a', we have 828 LONGITUDE. (»)-+(^)^ + and we find the erroneous longitude Hence the error by simple interpolation, commencing with the station -B, is dk'^ = — ^r'r'' ; and the error in the mean of the two longitudes is an error which disappears altogether when the intervals r and r" are equal. Since the voyages are of very nearly equal duration, it follows that by computing the longitude, as proposed by Struve, commencing alternately at the two stations, the final result will be free from the effect of any regular acceleration or retardation of the chronometers. Example. — From the " Expedition Chronom6trique" we take the following values for the chronometer "Ilauth 81,** being the combination next following after that given in the example of the preceding article, commencing now with the station jB, or Altona : At Altona {B\ t = May 26, 10*.72 6 = — 1* 14- 36'.77 " Pulkova {A\ a =z " 31, .00 a = + 7 9 .68 « Pulkova {A), t' = June 3, 5 .62 «' = + 7 19 .36 '' Altona (J?), r= " 7, 20 .52 6'= — 1 14 0.35 Here T = 4' 13*.28 = 4«'.553 6' — 6 = + 36*.42 t"= 4 14 .90 = 4 .621 a' — a=+ 9 .78 m = ^^'-^^ - ''- '"^ =^-'^ = + 2>.904 4.553 + 4.621 9.174 6 = — 1* 14- 36*.77 mr= + 13.22 (b) = — l 14 23.55 a = + 7 9 .58 ; = a — (6) = + 1 21 33 .13 The mean of this result and that of Art. 217 is ;i = 1* 21- 38'.02. BT OHRONOMSTERS. 829 219. Sdatwe weighi of the longitudes determined in different voyages by the same chronometer. — ^From the above it appears that the problem of finding the longitude by chronometers is one of interpolation. If the irregularities of the chronometer are regarded as accidental, the mean error of an interpolated value of the correction may be expressed by the formula"" -^'^vh + where r and z' have the same signification as in the preceding article, and e is the mean (accidental) error in a unit of time. The weight of such an interpolated value of the correction, and, therefore, also the weight of a value of the longitude deduced from it, is inversely proportional to the square of this error, and may, therefore, be expressed under the form where A is a constant arbitrarily taken for the whole expedition, BO as to give p convenient values, since it is only the relative weights of the different voyages which are in question. But if the chronometer variations are no longer accidental, but follow some law though unknown, a special investigation may serve to give empirically a more suitable expression of the weight than the above. Thus, according to Struve's investiga- tions in the case of certain clocks, the weight of an interpolated value of the correction for these clocks could be well expressed by the formulaf But even this expression he found could not be generally applied ; and he finally adopted the following form for the chronometric expedition : p = ^ (389) in which T is the duration of an entire voyage, including the ♦ See Vol. II., "Chronometer." t Expidition Chron., p. 102. 380 LOKGITUDK. time of rest at one of the stations, r, t'' are the travelling times of the voyage to and from a station, and £* is an arbitraiy constant. Although this is but an empirical formula, it represents well the several conditions of the problem. For^Jirsiy the weight of a resulting longitude must decrease as the length of the voyage increases ; and, second, it must become greater as the difterence between the tvvo travelling times r, t'' decreases, since (as is shown in Vol. EL., " Chronometer'*) an interpolated value of a clock correction is probably most in error for the middle time between the two instants at which its corrections are given. 220. Combination of results obtained by the same chronometer^ according to their weights. — ^Let >l', X", V" be the several values of the longitude found by the same chronometer, according to the method of Arts. 217 and 218 ; and p', />", />'" their weights by formula (389) (or any other formula which may be found to represent the actual condition of the voyages) ; then, according to the method of least squares, the most probable value of the longitude by this chronometer is L = y^- + i>-^- + J>-^- + (390) V' + f + /" + and if the diflference between this value and each particular value be found, putting n = the number of values of A, c =r the mean error of L, r = the probable error of L, then we shall have \(n-l)| r = 0.6745 c (391) [P] where [p] denotes the sum of /?', p^', &c., and \_pvv] the sum of p'v'v', ;/'«;"?;", &c. 221. Combination of the results obtained by different chronometers, according to their weights. — The weights of the results by ditlercut BY CHRONOMETERS. 881 chronometers are inversely proportional to the squares of their mean errors. The weight P of a longitude L will, therefore, be expressed generally by cc « in which k is arbitrary. For simplicitj', we may assume k = 1, and then by the above value of e we shall have p _ ("-1)^ (392) K, then, i', L"^ U" are the values found by the several chronometers by (390), P', P'', P''' their weights by (392), the most probable final value of the longitude is _ rV + T-L^^-^T"^L-^-^ ° P' + P" + P'" + ^ Then, putting i'-io=^'i X"~Xo=^'S i'"-io=^'" *c. iV = the number of values of X, E = the mean error of L^ B = the probable error ofIf^ 01 we have E = J_i^2n_ B = 0.6745 -B (894) 222. I propose to illustrate the preceding formulae by applying them to two chronometers of Struvb's expedition, namely, "Dent 1774" and "Ilauth 31." In the following table the longitudes found by beginning at Pulkova are marked P, those found by beginning at Altona are marked -4, and the numeral accent denotes the number of the voyage. The weights p in the second column are as given by Struve, who computed them by the formula (389), taking K= 34560 (the intervals T, r, r" being in hours), which is a convenient value, as it makes the weight of a voyage of nearly mean duration equal to unity ; namely, for T= 288*, t = t' = 120*. If we express T, r, r'', in days, we take (24)' 882 LONGITUBE. and we shall have Struve's values o(p by the formula P 60 Tv/rr" (395) Thus, for the first voyage, we have, from the data in the example of Art. 217, T=f"—t== 11' 2*.46 = 1K103 r = 6'.047 t"= 4'.663 whence, by (395), P = 60 11.103 >/(5.047 X 4.653) = 1.13 The values of i' and i" are found by (890). In applying this formula, it is not necessary to multiply the entire longitudes by their weights, but only those figures which difier in the several values. Thus, by "Dent 1774" we have Z' 1* 21- 30- -^ 2'-51 X 1.10 + 2'.83 X 102 + 2'.09 X 1.14 + Ac. "*" 1.10 + 1.02 + 1.14 + 4c. 1* 21- 30* -4- 2* 46 = 1* 21- 30* + 2'.46 Weight. P p» 1.18 A» 1.06 Ptt 1.10 A»» 1.02 pm 1.14 A"» 1.06 pir 1.19 A*'^ 0.96 P'^ 1.09 A' 0.80 pn 1.00 A^ 1.10 prll 1.20 AvU 1.09 prlU 0.76 ATlU 0.41 Longitudef by Chronometer Dent 1774. 1* 21"» 32'. 51 82.83 32.09 82.25 81 .69 32.77 82.79 82.54 32.94 81.93 82.84 82.95 81.86 83.77 4-0'.05 -f 0.37 — 0.37 — 0.21 — 0.77 -h 0.81 -f 0.88 -f-0.08 -f 0.48 — 0.53 — 0.12 + 0.49 — .60 -f 1 .31 ptv 0.003 0.140 0.156 0.046 0.706 0.092 0.119 0.005 0.230 0.309 0.017 0.262 0.274 0.704 Z' = l»21'"32«.46 n = 14 [pvv^ = 3.063 [p] = 18.91 P'= f'^- 13 X 18.91 3.063 .6745 = 59.04 = ±: 0'.09 LoDgitudet by Chronometer HanMSl. 1* 21"* 32«.91 33.18 83.36 33.12 82.55 81.56 82.70 84.16 82.23 31.65 83.38 31.97 38.16 31.78 80.92 4- ©-.SO -\- 0.52 -f 0.75 -f- 0.51 — 0.06 — 1.06 -f 0.09 -f 1.55 — 0.38 — 0.96 -f 0.77 — 0.64 4-0.55 — 0.88 — 1.69 0.102 0.287 0.619 0.265 0.004 1.158 0.010 2.8M 0.157 0.787 0.59S 0.451 0.863 0.761 2,171 Z"= 1* 21'" 32'.61 [/>rp] = 9.974 n = 15 [p] = 15.69 UXi6^9^22.^ 9.974 r" = .6745 v/P" O-.U BY CHRONOMETERS. 3S3 Combining these two results, we have, by (393), i. = P 21- 32. + 0--t6 X 59 + 0..61 X 22 _ ^^ ^l- 82. 601 * 59 + 22 with the probable error, by (894), B=± 0*.067 This agrees very nearly with the final result from the sixty-eight chronometers. 223. In the preceding method, the sea rate is inferred from two comparisons of the chronometer made at the same place before and after the voyages to and from the second place ; and the correction of the chronometer on the time of the first place at the instant when it is compared with the time of the second place is interpolated upon the theory that the rate has changed uniformly. This theory is insufficient when the temperature to which the chronometer is exposed is not constant during the two voyages, or nearly so. I shall, therefore, add the method of introducing the correction for temperature in cases where circumstances may seem to require it. According to the experience of M. Lieusson, the rate w of a chronometer at a given temperature & may be expressed by the formula (see Vol. IE., " Chronometer") m = w^ + A:(* — d^ — k't (396) in which &q is the temperature for which the balance is compen- sated, m^ the rate determined at that temperature at the epoch iz=Oy t being the time from this epoch for which the rate m is required, k the constant coefficient of temperature, and k' that of acceleration of the chronometer resulting from thickening of the oil or other gradual changes which are supposed to be pro- portional to the time. It is evident that, since every change of temperature produces an increase of tw, the term A:(i? — ^^y will not disappear even when the mean value of & is the same as i?^. It is necessary, therefore, to determine the sum of the effects of all the changes. Let us, therefore, determine the accumulated rate for a given period of time r. Let tWq be the rate at the middle of this period, in which case we have in the formula / = 0. A strict theory requires that 834 LONGITUDE. we should know the temperature at every instant ; but, in default of this, let us assume that the period r is divided into sufficiently small intervals, and that the temperature is obsen^ed in each. Let us suppose n equal intervals whose sum is r, and denote the observed values of ^ by tf <»>, m, tf (»> tf<->. The rate in the Ist interval is [m^ + k (^<" — #.)«] X — " 2d " K + A: (*<«>- d,)*] X^ in the nth interval is [m^ + k (^^ — 6^)^ X — and the accumulated rate in the time r is the sum of these quantities, where I^{9 — &qY denotes the sum of the n values of {& — &^. To make this expression exact, we should have an infinite number of infinitesimal intervals, or we must put - = rfr, and substitute the integral sign J for the summation symbol 2*: thus, the exact expression for the whole rate in the time r is m,r + kfJ(^^-{^,ydT (397) This integral cannot be found in general terms, since & cannot be expressed as a function of r ; but we can obtain an approxi- mate expression for it, as follows. Let ^^ be the mean of all the observed values of ^ ; then we have -n (* - *•)'= -n [(^I - ^) + (^ - ^l)]' in which ^^ — ^^ is constant, and, therefore, for n values we have -T^ («?i— ^'o)*— ^ (''i" ^o)'* Moreover, since tf^ is the mean of all the values of 1?, we have 2*^ (<? — <?,) = 0, and, consequently, also -r/2((?j-«?,)((>-t?i) = 2(«?i-t?,) -r.(i?-t?J = 0; and the'above expression becomes BT CHRONOMETERS. 835 bence, also, or, for an infinite value of w, Thus, the required integral depends upon the integral T (^ — ^i)* dvj which may be approximately found from the observed values of i^ by the theory of least squares. For, if we treat the values of i^ — tf 1 as the errors of the observed values of t?, and denote the mean error (according to the received acceptation of that term in the method of least squares) by €, we have e^^^Jtupi (398) n — 1 in which n is the actual number of observed values of tf . If we assume that a more extended series of values, or indeed an infi- nite series, would exhibit the same mean error (which will be the more nearly true the greater the number n), we assume the general relation in which N is any number. Hence, also, 2:^(^-^^).i. = r6« -ar— 1 N N and, making N infinite, S^i^-^ydr^r^ (899) Substituting this value, the fonnula (397) becomes or [m, + k {\ — d,)« + A'e«]r (400) from which it appears that vIq + h {9^ — ^^f + Are* is the mean rate in a unit of time for the intcr\'al r, m^ being the rate at the middle of the interval for a temperature t? = i?,,. For any subse- quent inten^al r', we must, according to (396), replace rw^ by m^ — k% i being the interval from the middle of r to the middle of r'. 836 LONGITUPE. Now, let us suppose that the chronometer correction is obtained by astronomical observations at the station Ay at the times T^ and T^, before starting upon the voyage, and again after reaching the station 5, at the times T^ and T^ these times being all reckoned at the same meridian. Let a^, a^ a,, a^ be the observed corrections, and put so that r and r" are the shore intervals and r' the sea intervaL Let the adopted epoch of the rate m^ be the middle of the sea interval r' ; then, by (400), with the correction k't, the accumu- lated rates in the three intervals are X + a,-^a,=:lm, +A:(^/ _ d,)« + A:e'« ] r' \ (401) in which T?p tf/, t?/' are the mean temperatures in the intervals T, r', r", and €, e', e^' are found by the formula (898). These three equations determine the three unknown quantities m^ Jf, and X. If we put / = ^^-^^^ - ^(^ - *o)'-*^' T we have, from the first and third equations, ^+1(^+t") which substituted in the second equation gives L If, however, we prefer to compute the approximate longitude without con- sidering the temperatures, and afterwards to correct for tempe> rature, we shall have TERRESTRIAL 8I0NALS. 887 These formulse apply to a voyage in either direction ; but in the case of a voyage from west to east they give X Avith the negative sign. The term jA'(r'' — r) r' in the first equation of (402) will not be rigorously obtained if the temperatures are neglected ; but it is usually an insensible term in practice, as r" and r are made as nearly equal as possible, and k' is always very small. In combining the results of different chronometers employed in the same voyage, the weight of each may be assigned accord- ing to the regularity of the chronometer as determined from its observed rates from day to day.* SECOND METHOD. — BY SIGNALS. 224. Terrestrial Signals. — If the two stations are so near to each other that a signal made at either, or at an intermediate station, can be observed at both, the time may be noted simultaneously by the clocks of the two stations, and the difference of longitude at once inferred. The signals may be the sudden disappearance or reappearance of a fixed light, or flashes of gunpowder, &;c. If the places are remote, they may be connected by interme- diate signals. For example : suppose four stations, Ay B, (7, -D, chosen from east to west, the first and last being the principal stations whose difference of longitude is required. At the in- termediate stations B, C let observers be stationed with good chronometers whose rates are known. Let signals be made at three x>oints intermediate between A and B, B and C, Cand Z), respectively. The signals must, by a preconcerted arrangement, be made successively, and so that the observers at the interme- diate stations may have their attention properly directed upon the appearance of the signal. If, then, at the first signal the observers at A and B have noted the times a and b; at the * Besides the papers already referred to, sec the Report of the Superintendent of the U. S. Coast Surrey for 1857, p. 814. Vol. L— 22 838 LONGITUDE. second signal the observers at ^ and Cthe times b' and c; at the third signal the observers at C and D the times c' and d; it is evident that the time at A when the third signal is made is a + (6'— i) + {c' — c\ at which instant the time at i> is d: hence the difference of longitude of A and D is >l = a + (6' — 6) + (c' — c) — d (403) and so on for any number of intermediate stations. It is re- quired of the intermediate chronometers only that they should give correctly the differences 6'— 6, c'— <?, for which parpose only their rates must be accurately known. The daily rates are obtained by a comparison of the instants of the signals on suc- cessive days. Small errors in the rates will be eliminated by making the signals both from west to east and from east to west, and taking the mean of the results. The intervals given by the intermediate chronometers shouldf of course, be reduced to sidereal intervals, if the clocks at the extreme stations are regulated to sidereal time. Example. — ^From the Description GSomStrique de la Frasi^ (Puissant). On the 25th of August, 1824, signals were observed between Pari^ and Sirasburg as follows: Paris. A 19* 6" 20'.3 Intermediate Stations. B 8* 49- 48'.2 8 54 10.8 9*16" 0'.2 9 30 37.8 Strasborg. D 19* 46- 51-.4 The correction of the Paris clock on Paris sidereal time was — 36*.2 ; that of the Strasburg clock on Strasburg sidereal time was — 27'.7. The chronometers at B and Cwere regulated to mean time, and their daily rates were so small as not to be sensible in the short intervals which occurred. "We have U—hz Mean interval Eed. to Bid. int. = Sid. interval =19 3 .3 4« 22*.6 14 37.6 19 0.2 + 3.1 CELESTIAL SIGNALS. 8S9 Paris clock 19* 6* 20*.3 Strasburg clock 19* 46* 51'.4 Correction — 36 .2 Correction — 27 .7 Paris sid. time 19 6 44 .1 Strasburg sid. time 19 46 23.7 Sid. interval +19 3 .3 Paris sid. time of the 'j last signal } ^^ ^4 47 .4 Strasburg do. 19 46 23.7 X= 0»21*36'.3 In the survey of the boundary between the United States and Mexico, Major W. H. Emory, in 1852, employed flashes of gun- powder as signals in determining the dift*. of long, of Frontera and San Elciario."*" The signals may be given by the heliotrope of Gauss, by which an imago of the sun is reflected constantly in a given direction towards the distant observer. Either the sudden eclipse of the light, or its reappearance, may be taken as the signal; the eclipse is usually preferred. Among the methods by terrestrial signals may be included that in which the signal is given by means of an electro-tele- graphic wire connecting the two stations; but this important and exceedingly accurate method will be separately considered below. 225. Celestial Signals. — Certain celestial phenomena which are visible at the same absolute instant by observers in various parts of the globe, may be used instead of the terrestrial signals of the preceding article : among these we may note — a. The bursting of a meteor, and the appearance or disappear- ance of a shooting star. — The difficulty of identifying these objects at remote stations prevents the extended use of this method. 6. The instant of beginning or ending of an eclipse of the moon. — This instant, however, cannot be accurately observ^ed, on account of the imperfect definition of the earth's shadow. A rude approximation to the difference of longitude is all that can be expected by this method. e. The eclipses of Jupiter's satellites by the shadow of that planet. — The Greenwich times of the disappearance of each * Proceedings of 8th Meeting of Am. Association, p. 64. satellite, and of its reappearance, are accurately given i^ Ephemeris : so that an observer who has noted one of these ]ilienoinenft basjonly to take the difFerenoe between this observed local time of Its occurrence and the Greenwich time gi\-en in the E]>hemoris, to have his absolute longitude. With telescopes of dili'crent powers, however, the instant of a satellite's disappear- ance must evidently vary, since the eclipse of the satellite take» place gradually, and the more powerful the telescope the longi-r will it continue to show the satellite. If the disappearance and reappearance are both observed M'ith the eorae telescope, the mean of the results obtained will be nearly free from this error. The first satellite is to be preferred, as its eclipses occur mor* frequently and also more suddenly. Observers who wish to deduce their difference of longitude by these eclipses should use telescopes of the same power, and obseiwe under the sauie atmospheric conditions, as nearly as possible. But in no case can extreme precision bo attained by this method. il. The occiiUations of Jupiter's satellites by the body of tlis planet. — The approximate Greenwich times of the disappearance behind the disc, and tlie reappearance of each satellite, are given in the Epheineris. These predicted times servo only to enable the observers to direct their attention to the phenomenon at the proper moment.- c. The transits of the satcUtteB over Jupiter's disc. — The ap- proximate Greenwich times of "ingress" and '•egress," or the first and last instants when the satellite appears projected on the planet's disc, are given in the Ephemeris, /. The transits of the shadows of the satellites over Jnpiter'n disc. — The Greenwich times of "ingress" and "egress" of the shadow are also approximately given in the Ephemeris. Among the celestial signals we may include also eclipsea of the sun, or occultationa of stars and planets by the moon, or, in general, the arrival of the moon at any given position in the heavens; but, in consequence of the moon's parnllax, these eclipses and occultationa do not occur at the same ab^lute in- stant for all observei-s, and, in general, the moon's appHrrnl position in the heavens is affected by both parallax and refrac- tion. The methods of emplojing tliese phenomena as ei^als^ therefore, involve special computations, and will be hereafter treated of. See the general theorj' of ecUpnes, and the method of lunar distances. BY TUE ELECTRIC TELEGRAPH. 341 THIRD METHOD. — BY THE ELECTRIC TELEGRAPH. 226. It is evident that the clocks at tri^o stations, A and B^ may be compared by means of signals communicated through ail electro-telegraphic wire which connects the stations. Sup- pose at a time T by the clock at A, a signal is made which is perceived at B at the time T' by the clock at that station. Let ATand a 7^' be the clock corrections on the times at these sta- tions respectively (both being solar or both sidereal). Let x be the time required by the electric current to pass over the wire ; then, A being the more easterly station, we have the difference of longitude X by the formula >l = (T + aT) — (r'+ aT') + a- = A, + a: Since x is unknown, we must endeavor to eliminate it. For this purpose, let a signal be made at B at the clock time T'\ which is perceived at A at the clock time T'" ; then we have In these formulae X^ and ^ denote the approximate values of the difference of longitude, found by signals east-west and west-east respectively, when the transmission time x is disregarded; and the true value is Such is the simple and obvious application of the telegraph to the determination of longitudes; but the degree of accuracy of the result depends greatly — more than at first appears — upon the manner in which the signals are communicated and received. Suppose the observer at A taps upon a signal key* at an exact second by his clock, thereby producing an audible click of the armature of the electro-magnet at B, The observer at B may not only determine the nearest second by his clock when he hears this click, but may also estimate the fraction of a second; and it would seem that we ought in this way to be able to deter- mine a longitude within one-tenth of a second. But, before even this degree of accuracy can be secured, we have yet to eliminate, or reduce to a minimum, the following sources of error: * See VoL IT., ** Chronograph," for the detdls of the apparatus here alluded to. 6i2 LOSGITUDE. iBt. Tho persona] error of tlio observer who gives the signat" 2d. The pci-9onal error of the observer who rcteives tho signal und ostimalea tho frncUon of a second bj- the ear; Sd. Tho small fraction of time required to complete the galvanic circnit after tho finger touches tho signal key; 4th. Tho armature time, or the time reqnired hy the arinatare at the station where the signal is received, to move through tho space in whieli it plays, and to give the audible click; 5th. Tho errors of tho BUppoeed dock corrections, which involve errors of observation, and errois in the right asceiieionut of the stars employed. For the means of contending successfully ■with these sources of error we are indebted to our Coast Burvey, which, under the superintendence of Prof. Bache, not only called into existence the chronogmphic instruments, but has given ua the tnoet effi- cient method of using them. The "method of star signals," »a it is called, was originally suggested by the distinguished Juttro- nomer Mr. S. C. "Walker, but its full development in the fonu now employed in the Coast Survey is due to Dr. B. A. Gould. 227. Method of Star S^als.—The difference of longitude Imn tween the two stations is merely- the time required by a »t»r to pass from one meridian to the other, and this interx-al may be measured by means of a single clock placed at either stutioii," but in the main galvanic circuit extending trom one stntioa to the other. Two chronographs, one at each station, are also in the circuit, and, when the wires are suitably connected, the clock seconds are recorcied upon both. A good transit instranient is carefully mounted at each station. Wlien the star enters the field of the transit instrument at A (the eastern station), the observer, by a preconcerted signal with his signal key, gives notice to the assistants at both A and B, who at once set the chronographs in motion, and the clock then records its seconds upon both, Tlie instants of tlic star* tran- sits over the several threads of the reticule are also recorded upon both chrouographs by the tajia of the observer upon his signal key. When tlie star has passed all the threads, the ob- * Tbe olock ma;, intlnd. be Ml bii; pUce which la in telegrapUa Mmaadioa vU the (wo iMlioDi wboH differenoe at lonplude is to be IMukL BT THE ELECTRIC TELEGRAPH. 843 server indicates it by another preconcerted signal, the chrono- graphs are stopped, and the record is suitably marked with date, name of the star, and place of observation, to be subsequently identified and read off accurately by a scale. Wlien the star arrives at the meridian of -B, the transit is recorded in the same manner upon both chronographs. Suitable observations having been made by each observer to determine the errors of his transit instrument and the rate of the clock, let us put T^ = the mean of the clock times of the eastern transit of the star over all the threads, as read from the chrono- graph at Af T, = the same, as read from the chronograph at B, 7\' = the mean of the clock times of the western transit of the star over all the threads, as read from the chrono- graph at A, TJ =z the same as read from the chronograph at By 6, e' = the personal equations of the observers at A and B respectively, t,t'= the corrections of T^ and 2\' (or of T.and T^') for the state of the transit instruments at A and B, or the respective "reductions to the meridian" (Vol. II., Transit Inst.), dT= the correction for clock rate in the interval T/ — 2\, X = the transmission time of the electric current between A and By X = the difference of longitude ; then it is easily seen that we have, from the chronographic records at J., >l = r/ + ^r+ t' + e' — a; — (2; + T + c) and from the chronographic records at -B, and the mean of these values is ^ = [K^i'+^«0 + ^]-[Kri+2;) + r] + Jr+e'-e (404) which we may briefly express thus : 344 LONGITUDE. in which X^= the approximate difference of longitade found by the exchange of star signals, when the personal eqaations of the observers are neglected. This equation would be final if e' — e^ or the relative personal equation of the observers, were known : however, if the observew now exchange stations and repeat the above process, we shall have, provided the relative personal equation is constant, in which ^ is the approximate difference of longitude found aa before ; and hence the final value is I have not here introduced any consideration of the armature time, because it affects clock signals and star signals in the same manner; and therefore the time read from the chronographic fillet or sheet is the same as if the armature acted instanta- neously.* It is necessarj', however, that this time should be constant from the first observation at the first station to the last observation at the second, and therefore it is important tliat no changes Bhould be made in the adjustments of the apparatus during the interval. As the observer has only to tap the transits of the star over the threads, the latter may be placed very close together. The reticules prepared by Mr. AV. AVirdemanx for the Coast Sur\-iy have generally contained twenty-five threads, in groups or '* tal- lies" of five, the eipiatorial intenals between the threads, of a group being 2'.5, and those between the grou])8 o' ; with an ad- ditional thread on each side at the distance of lO* for use in olv servations by '' eye and ear.'' Except when clouds inten-ene and render it necessarv to take whatever threads mav be avail- able, only the three middle tallies, or fifteen threads, are used. The use of more has been found to add less to the accuracy of a ♦ Dr. B. A. GorLD thinks that the armnture time raries with the fltrcof^h of the battery and the distance (and consequent -weakness) of the signal; being thus liable to be confounded with the transmission time. The effect upon the difference of longitude wiU be inappreciable if the batteries are maintained at nearly the strength. BT THE ELSCTBIC TELEGRAPH. &15 determination than is lost in consequence of the greater fatigue from concentrating the attention for nearly twice as long. A large number of stars may thus be ob8er\'ed on the same night ; and it will be well to record half of them by the clock at one station, and the other half by the clock at the other station, upon the general principle of varying the circumstances under which several determinations are made, whenever practi- cable, without a sacrifice of the integrity of the method. For this reason, abo, the transit instruments should be reversed during a night's work at least once, an equal number of stars being observed in each position, whereby the results will be freed from any undetermined errors of collimation and inequality of pivots. Before and after the exchange of the star signals, each observer should take at least two circumpolar stars to determine the instrumental constants upon which r and r' depend. This part of the work must be carried out with the greatest precision, employing only standard stars, as the errors of r and r' come directly into the difference of longitude. The right ascensions of the "signal stars" do not enter into the computation, and the result is, therefore, wholly free from any error in their tabular places : hence any of the stars of the larger catalogues may be used as signal stars, and it will always be possible to select a sufficient number which culminate at moderate zenith distances at both stations, (unless the difference of latitude is unusually great), so that instrumental errors will have the minimum effect. A single night's work, however, is not to be regarded as con- clusive, although a large number of stars may have been ob- sen-ed and the results appear verj' accordant; for experience shows that there are always errors which are constant, or nearly so, for the same night, and which do not appear to be represented in the corrections computed and applied. Their existence is proved when the mean results of different nights are compared. Moreover, it is necessary to interchange the observers in order to eliminate their personal equations. The rule of the Coast Survey has been that when fifty stars have been exchanged on not less than three nights, the observers exchange stations, and fifty stars are again exchanged on not less than three nights. The observers should also meet and determine their relative personal equation, if possible, before and after each series, as it may prove that this equation is not absolutely constant. 846 LONGITUDE. Before entering upon a series of star signals, each observer will be provided with a list of the stars to be employed. The preparation of this list requires a knowledge of the approximate difference of longitude in order that the stars may be so selected that transits at the two stations may not occur simultaneously. Example. — For the purpose of finding the difference of longi- tude between the Seaton Station of the U. S. Coast Survey and Raleigh, a list of stars was prepared, from which I extract the following for illustration. The latitudes are Seaton Station (Washington) ^ = + 38* 53'.4 Raleigh " (North Carolina) ^ = + 35 47 .0 and Raleigh is assumed to be west from Washington 6* 30*. Seaton sidereal 8tAr. Mag. a 6 timeof Raleigk transit. No. 5036 B.A.C. 3 15* 9- 36* + 33* 52' 15* 16- 6* 5084 4.3 18 58 37 54 25 28 5131 H 27 2 31 51 33 32 5192 5 36 35 26 46 43 5 5259 5 45 43 36 7 52 13 5322 41 55 59 23 12 16 2 29 5388 5 16 4 9 45 19 10 39 5463 3.4 15 21 46 40 21 51 The following table contains the observations made on one of these stars at the above-named stations by the U. S. Coast Suney telegraphic party in 1853, April 28, under the direction of Dr. B. A. Gould. In this table " Lamp W.*' expresses the position of the rotation axes of the transit instruments. The 1st column contains the sym- bols bv which the fifteen threads of the three middle tallies were denoted; the 2d column, the times of tnmsit of the star over each thread at Seaton, as read from the chronographs at Seaton; the 3d column, the times of these transits as read from the chro- nographs at Raleigh ; the 4th column, the mean of the 2d and 3d columns ; the 5th column, the reduction of each thread to the mean of all, computed from the known equatorial inter\'al8 of the threads ; the 6th column, the time of the starts transit over BY THE ELECTRIC TELEQBAPH. »47 liie mean of the threads, being the algebraic sum of the numbers in the 4th and 5th columns ; and the remaining columns, the Baleigh observaUons similarly recorded and reduced. SBATON— RALEIOU, 1863 AprU 28. BUr No. t!M B. A. a e—taaOb^ UmpW. Rmkiiili 01m. UdipW. r,+ T, r,'+r,' """**■ r. r. Mhui Hcd. '■' r," Heu. Red. 2 ^ 3J:m 3«'.INI 3TM + ZS'.« 3.47 11' 00 IKOO + »•« se.*5 4t--n 41 ja SJ.I1 i4^i« u !m 22.19 38 J9 W.l^ 19 JW 1T.W1 30.63 c. IS .71 3.ia ao !b8 jo .79 c; 6U .7(1 » .70 iO J3 ».T1 «-T-" a .80 1J.70 3S.U I^ Bin 80.1«».o» ■ 32 i< S3,3<!33.2i 3. IS M.1S ^ 3.ffl 3.30 IC 6> M .4U 3« .»> + .071 34 Ai 5 O.JO f3,«] ~ 3 .lo' 3« .11 ». 3.30 «^i«^ 0J.|3«.M E. 1«JU ia.03 G.M 11 M 3.44 sd^:SS-S 12 .76'[36 .17) K, Has 15 Tm 3vIS UJO 3a .42 E, 1S.W 3.47 K ±1S.,\ 3^8 M ,73|.S8 .80 M ,67 Z-J 20 SS .47 K 38 >0.2S !7U 2j ;ui 3 32 3.08lai>sl a.08 » jel M ,70 "*» ^3.3K ««-- »«| The numbers in the last column for each station would be equal if the observations and chronographic apparatus were perfect ; and by carrying them out thus individually we can estimate their accuracy. The numbers [3.67] at Seaton and [36.17] at Raleigh are rejected by the application of Peirce'b Criterion (see Ap- pendix, Method of Least Squares), and the given means are found from the remaining numbers. The corrections of the transit instruments for this star (i= + 86''6'.9)were for the Seaton instrument, t = — 0'.028 " " Raleigh " r" ^ — .193 The rate of the clock was iiisensibte in the brief interval r/ — T. Hence, neglecting the personal equations of the ob- Ber\-ers, the difference of longitude is found as follows: J(r,'+ r;) + r'^lfl'52-36'.342 j(r,+ r.)+r = 15 46 3.36A i,= 6 32.978 In this manner seven other stars w night, and the results were as follows : ■ observed on the same 348 LONOrrUDE. Star ^ Diff. from memn 5036 B. A. C. 6* 33-.03 ■j-O-M 5084 33.09 + 0.10 5131 « 32.91 — 0.08 5192 " 33.00 + 0.01 5259 " 32.98 — O.Ol 5322 « 33.00 + 0.01 5388 " 33.02 + 0.03 5463 « 32.91 — 0.08 Moan A, = 6 32 .99 From the residuals i?, we deduce the mean error of a sinsrle determination by one star, ■=V(;S)=sl(-T)— and hence the mean error of the value 6"* 32*.99 is 0v06 1/8 = ± 0-.02 But this error will be somewhat increased by those errors of the instruments which are constant for the night, and not represented in r and r', and by the errors of the personal equations yet to bo applied. Moreover, a greater number of determinations should be compared, in order to arrive at a just evaluation of the mean error. 228. VcloeiO/ of the galvanic current — Recurring to the equations of p. 343, we find, by taking the difforence between the values of / given by the chronograi)hic records at the two stations, x=\{T;-T:)+\{T,^T,) If the clock is at the eastern stiition (yl), the time T^ will not differ from TJ, except in consequence of irregularities in the chronographs and errors in reading them, and therefore we should find z solely from the times 2\' and 2\\ or x=\{T^^T:) (405) BY THE ELECTRIC TELEGRAPH. 349 In like manner, if the clock is at the western station, we find x by the formula a: = 1(2; -TO Thus, in general, the transmission time will be deduced by com- paring the records of the star signals made at one station when the clock is at the other station. In the above example, the clock was at Washington, and hence, from the record of the transit at Raleigh, we have fourteen values of T^— 7i'= 2x, as follows: + 0'.08 + 0'.08 + .05 + 00 + .09 + .23 + .03 — .03 + .14 + .09 + .09 + .13 + .10 + .00 That these are not merely accidental residuals is shown by the permanence of sign, with the single exception in the case of the eleventh observation. The discrepancies between them indicate accidental variations in the chronographs, combined with errors in reading off the record. Taking the mean, as elimi- nating to a certain extent these errors, we have 2x = 0*.077 X =. 0*.0385 From this value of x and the distance of the stations we can deduce the velocity per second of the galvanic current. In the present instance, the length of the wdre was very nearly 300 miles, and, if the above single observation could be depended upon, we should have, velocity per second = -^tt^^ttj^ = 7792 miles, which is doubtless too small. The velocity thus found, however, appears to depend upon the intensity of the current,* as has been shown by varying the battery power on different nights. It has also been found that the velocities dctennined from signals made at the east and west stations differed, and that this difference was api)arently depend- * It depends also upon the sectional area, molecular structure, and, of course, materUl, of the wire. 850 LONGITUDE. eiit upon the strength of the batteries ; the velocities from signak east-west and signals west-east coming out more and more nearly equal as the strength of the batteries was increased. See Dr. Gould's Report on telegraphic determinations of differ- ences of longitude, in the Report of the Superintendent of the U. S. Coast Survey for 1857, Appendix No. 27. FOURTH METHOD. — BY MOON CULMINATIONS. 229. The moon*s motion in right ascension is so rapid that the change in this element while the moon is passing from one meridian to another may be used to determine the difference of longitude. Its right ascension at the instant of its meridian transit is most accurately found by means of the interval of sidereal time between this transit and that of a neighboring well- known star. For this puq^osc, therefore, the Ephemerides con- tain a list of moon-culminating stars, which are selected for each day so that at least four of them are given, the mean of whose declinations is nearly the same as that of the moon on that day, and, generally, so that two precede and two follow the moon. The Ephemerides also contain the right ascension of the moon*8 bright limb for each culmination, botli upper and lower, and the variation of this right ascension in one hour of longitude, — i.e. the variation during the interval between the moon's transits over two meridians whose difference of longitude is one hour. This variation is not uniform, and its value is given for the instant of the passage over the meridian of the Ephemeris. These quantities facilitate the reduction of corresponding obser- vations, as will be seen below. 230. As to the observation, let t^, d' = the sidereal times of the culmination of the moon's limb and the star, respectively, corrected for all tho known errors of the transit instramont, and for clock rate, a, a' = the right ascensions of the moon's limb and the star at the instants of transit; then we evidently have a==a' + * — *' (406) BY MOON CULMINATIONS. 851 The star and the moon being nearly m the same parallel, the instrumental errors which aftect i? also affect i?' by nearly the same quantity. We should not, however, for this reason omit to apply all the corrections for hwum instrumental errors, since by this omission we should introduce an error in the longitude precisely equal to the uncorrected error of the instrument. For if the instrumental error produces the error z in the time of the star's transit, the effect is the same as if the instrument were perfectly mounted in a meridian whose longitude west of the place of observation is equal to z ; but the sidereal time required by the moon to describe this interval z is equal to 2 + the increase of the moon*s right ascension in this interval. Hence the longitude found, by the methods hereafter given, would be in error by the quantity z. 231. If the lunar tables were perfectly accurate, the true longitude given by the observation would be found at once by comparing the observed right ascension with that of the Ephe- meris. There are two methods of avoiding or eliminating the errors of the Ephemeris. In the first, which has heretofore been exclusively followed, the observation is compared with a corre- sponding one on the same day at the first meridian, or at some meridian the longitude of which is well established. In this method the increase of the right ascension in passing from one meridian to the other is directly observed, and the error of the Ephemeris on the day of observation is consequently avoided ; but observations at the unknown meridian are frequently ren- dered useless by a failure to obtain the correspondmg observa- tion at the first meridian. In the second method, proposed by Professor Peircb, the Ephemeris is first corrected by means of all the observations taken at the fixed observatories during the semi-lunation within which the ob8er\'ation for longitude falls. The corrected Ephe- meris then takes the place of the corresponding ob8er\'ation, and is even better than the single corresponding observation, since it has been corrected by means of all the observations at the fixed obsen^atories during the semi-lunation. I shall consider first the method of reducing corresponding observations. 852 LONGITUDE. 232. Correspo)idmg observations at places whose difference of longi- tude is less than two hours. — At each place the true sidereal times of transit of the moon-culminating stars and of the moon*8 bright limb are to be obtained with all possible precision : from these, according to the fonnula (406), will follow the right as- cension of the moon's limb at the instants of transit over the two meridians, taking in each case the mean value found from all the stars obscrv^ed. Put Xj, Xg = the approximate or assumed longitudes, X = the true diiferenco of longitude, ttj, ttj, = the observed right ascensions of the moon's bright limb at L^ and i, respectively, Ji^ = the variation of the R A. of the moon's limb for 1* of longitude while passing from L^ to i, ; then we have ;=.!^' (407) in which, a, — ttj and IIq being both expressed in seconds, k will be in hours and decimal parts. When the difterence of longitude is less than two hours, it is found to be sufficiently accurate to regard Jl^ as constant, provided we employ its value for the middle longitude Jj^ =1 h{L^ -p ij), found by interpolation from the values in the Ei)hemeris, having regard to second differences. Example. — The following]: observations were made, ^^av 15, 1851, at Santiago, Chili, by the U. S. Astronomical Kxpeditiou under Lieut. Gilliss, and at Philadelphia, by Prof. Kendall: Object. Santiago sid. time. ' PhiUd'a ttid. time. | tS' Librae 15* 40- 8-.87 , 15» 45- 22v:W Moon II Limb ' 1() 21 30.84 : 10 21 39.11 B. A. C. 5570 I 10 33 40 .12 | 10 32 5S M "We shall assume the longitudes from Greenwich to be, Philadelphia, L, ^ 5* 0- 39-.S5 Santiago, 7., r_-^ 4 42 19. the longitude of Philadelphia being that which results from the last chnaiometrir expeditions of the U. S. Coast rSur\-ey, and that of Santiago the value which Lieut. Gilliss at first assumed. BT MOON CULMINATIONS. 853 The apparent right ascensions of the stars on May 15, by the moon-cuhninating list in the Kautieal Almanac, were * Librae B. A. C. 5579 15* 45- 22'.59 16 32 59.20 "We have then at Philadelphia, by (406), ^ — ^ o'+i^— 1^ * Librae B. A. C. 5579 + 36- 16'.78 — 11 19.85 16* 21- 39'.37 16 21 39.35 and at Santiago : ^ Librae B. A. C. 5579 Hence Mean a^ = 16 21 39 .36 16 20 56.06 16 20 55.92 + 35 33.47 — 12 3 .28 Mean o, = 16 20 55 .99 I. — ttj = — 43'.37 We shall find H^ for the mean longitude i© = J (A + -^2) = 4*.86, by the interpolation formula (72), or in which, if we put n = A = n = 0.405 4*.86 "12"^ we have J = "("-^> = -0.120 2 and a' and b^ are found from the values of H in the Ephemeria as follows : May 15, L. C. 142'.56 , q. 92 2d dif. « 15, U. C. 143 .48 ^ ' __ 0-.28 + 0.64 [-0.85] « 16, L. C. 144.12 ^^o —0.41 '* 16, U. C. 144 .35 "^ whence H = 143'.48 a'= O'M b^=i(— 0'.28 — 0'.41) = — 0'.35 ff^ = 143'.48 + 0'.259 + 0'.042 = 143'.781 43 Q7 Vol. I.— 23 354 LONGITUDE. which is tlie longitude of Santiago from Philadelphia, Hence, if the longitude of Philadelphia is correct, we have Long, of Santiago == 4* 42* 33*.95 from Greenwich. 233. Corresponding observations at places whose difference of hngi- tude is greater than two hours, — Having found a^ and (Z, as in the preceding case, we employ in this case an indirect method of solution. For each assumed longitude we interpolate the right ascension of the moon's limb from the Moon Culminations in the Ephemeris to fourth differences. Let A^y A^ = the interpolated right ascensions of the moon's limb for the assumed longitudes L^ and X, respect- ively, If the correction of the Ephemeris on the given day is f, the true values of the right ascension for L^ and i, are A^ + t and A2 + c, the error of the Ephemeris being supposed to be sensibly constant for a few hours ; but their difference is ^A, + e)-{A, + e) = A,-A, so that the computed difference of right ascension is the same as if the Ephemeris were correct. If now the observed differ- ence ttj — tti is the same as this computed difference, the as- sumed difference of longitude, or i, — Z/^, is correct ;♦ but, if this is not the case, put r=K-a,)-(^-A) (m) and ^L = the correction of the uncertain longitude, which we will suppose to be i, then Y is the change of the right ascension while the moon is describing the small arc of longitude lL ; and for this small difference we may apply the solution of the preceding article, so that we have at once aX = -^ (in hours) (409) or ^L = r X (in seconds) (409*) * It should be obserred, however, that one of the assumed longitudes most be nearly correct, for it is evident that the same difference of right ascension wiU not exactly correspond to the same difference of longitude if we increase or deereast both longitudes bj the same quantity. BY MOON CULMINATIONS. 855 in which the value of H must be that which belongs to the uncertain meridian i,, or, more strictly, H must be taken for the mean longitude between Z^ and L^ + aX; but, as aZ/ is generally very small, great precision in If is here superfluous. However, if in any case aZ/ is large, we can first find H for the meridian i„ and with this value an approximate value of aX; then, interpolating J for the meridian i, + J aZ/, a more correct value of aX will be found.* Example. — The following observations were made May 15, 1851, at Santiago and Greenwich : Object. Santiago. Qreenwich. ^Librae 15M6-' 3'.37 15M5* 22'.37 Moon II Limb 16 21 36.84 16 9 39.41 B.A.C. 5579 16 83 40.12 16 32 59.17 We assume here, as in the preceding example, for Santiago i, = 4* 42* 19*, and for Greenwich we have L^ = 0. The places of the stars being as in the preceding article, we find for Greenwich, a^ = 16* 9* 39'.54 Santiago, o, = 16 20 55 .99 tt, — ttj= 11 16.45 The computed right ascension for Greenwich is in this case simply that given in the Ephemeris for May 15 ; the increase to the meridian 4* 42* 19*. has been found in our example of in- terpolation, Art 71, to be A^'-A^ = ll- 15'.84 and hence r = + 0'.6i We find, moreover, for the longitude 4* 42" lO", H = 143'.77 whence A£=-rO-.61X^^ = + 16-.28 By these observations we have, therefore. Longitude of Santiago = 4* 42- 34*.28 * This method of reducing moon culminations was developed by Walkkb, Trant* oetUmt of the American PhUotophieal Society^ new series, Vol. V. 356 LONGITUDE. 234. Reduction of moon culminations by the hourly JSphenieris. — The method of reduction given in the preceding aitiele is per- fectly exact ; but the interpolation of the moon's place to fourth difterences is laborious. The hourly Ephemeris, however, requires the use of second differences only. The sidereal time of the transit of the moon's centre at the meridian ij is = the observed right ascension of the centre = a^. K then we put T, = the mean Greenwich time corresponding to o, as found by the hourly Ephemeris, 0j = the Greenwich sidereal time corresponding to T^, we have at once, if the Ephemeris is correct, A = e, - «i (410) This, indeed, was one of the earliest methods proposed, but was abandoned on account of the imperfection of the Ephemeris. The substitution of corresponding observations, however, does not require a departure from this simple process ; for we shall have in the same manner, from the observations made at another meridian (which may be the meridian of the Ephemeris), i, = e, — tta and hence ; = i, - i. = (0, - e.) - (a, - a,) (411) and it is evident that the difference {Q^ — ©<,) of the Greenwich times will be connect, although the absolute right ascension of the Ephemeris is in error, provided the hourly motion is correct. The correctness of the hourly motion must be assumed in all methods of reducing moon culminations ; and in the present state of the lunar theorv there can be no error in it which can be sensible in the time required by the moon to pass from one meridian to another. In this method a is the rischt ascension of the moon's centre at the instant of the transit of the centre ; which mav be de- duced from the time of transit of the limb bv addino: or sub- tractingthc '' sidereal time of semidiameter passing the meridian,'* given in the table of moon culminations in the Ephemeris.* To find 7^1 corresponding to ttp we may proceed as in Art. 64, * If we wish to be altogether independent of the moon-culminating table, we can compute the sidereal time of semidiameter passing the meridian by the formula (we Vol. II., Transit Instrument), BY MOOfN CULMINATIONS. 857 or as follows: Let T^ and T^-\- 1* be the two Greenwich hours between which a^ falls, and put Ao = the increase of right ascension in 1"* of mean time at the time T^, da = the increase of Aa in 1*, oq = the right ascension of the Ephemeris at the hour T^, then, by the method of interpolation by second differences, we have 12 3600 J\ 60 / in which the interval 7\ — T^ is supposed to be expressed in seconds. This gives ^ 2 3600 and in the second member an approximate value of 7^ may be used, deduced from the local time of the observation and an approximate longitude. A still more convenient form, which dispenses with finding an approximate value of 7\, is obtained as follows : Put then we have 16(1 — A) cos d in which S = the moon^s semidiameter, X = the increase of the moon's right ascen- sion in one sidereal second, and 6 = the moon's declination, which are to be taken for the Greenwich time of the observation, approximately known from the local time and the approximate longitude. Or we may apply to the sidereal time (= t^j) of the transit of the limb the quantity 15 cos 6 and the resulting ai= -^^ dtz ^ S see 6 will be the right ascension of the moon's centre at the local sidereal time i^j. We then find the Greenwich time Oj corre- sponding to Oi as in the text, and we have 358 LONGITUDE. 60 (g, ~ »o) X = Aa \ ^ 7200 A» / 01', with sufficient accuracy, \ 7200*A»/ a: = Aa Putting then Att 7200 Aa we have, very nearly, x = x' + a:" (413) As a practical rule for the computer, we may observe that x" will be a positive quantity when Aa is decreasing, and negative when Aa is increasing. The method of this article will be found particularly conve- nient when the observation is compared directly with the Ephemeris, the latter being corrected by the following process. See page 362. 235. Peirce's method of correcting the Ephemeris,* — The accuracy of the longitude found by a moon culmination depends upon that of the observed difference of right ascension. Wlien this difference is obtained from two corresponding observations, if the probable errors of the observed right ascensions at the two meridians are e^ and e,, the probable error of the difference will be = V{e^ + fij*)- [Appendix]. But if instead of an actual ob- servation at Z/, we had a perfect Ephemeris, or 6,-0, the probable error of the observed difference would be reduced toe,; and if we have an Ephemeris the probable error of which is less than that of an observation, the error of the obser^'ed difference is reduced. At the same time, we shall gain the additional advantage that every observation taken at the meridian whose longitude is required will become available, even when no corre- sponding obscn-ation has been taken on the same day; and * Report of the Superintendent of the U. S. Coast Surrey for 1864, Appeadii, p. 116» BY MOQN CULMINATIONS. 859 experience has shown that, when we depend on corresponding observations alone, about one-third of the observations are lost The defects of the lunar theory, according to Peirce, are involved in several terms which for each lunation may be principally combined into two, of which one is constant and the other has a period of about half a lunation, and he finds that for all practical purposes we may put the correction of the Ephemeris for each semi-lunation under the form jr=A + Bt+ O" (414) in which -4, J5, and G are constants to be determined from the observations made at the principal observatories during the semi-lunation, and t denotes the time reckoned from any assumed epoch, which it will be convenient to take near the mean of the observations. The value of t is expressed in days ; and small fractions of a day may be neglected. Let a^y Oji, Og, &c. = the right ascension observed at any observa- tory at the dates f^ ^,, t^, &c., from the assumed epoch, a/,tt/,tt,',&c. == the right ascension at the same instant found from the Ephemeris, and put then rij, n^ n,, &c. are the corrections which (according to the observations) the Ephemeris requires on the given dates, and hence we have the equations of condition A + Bt^+ a* — Mj = A + Bt,+ a,'^n, = &c. In order to eliminate constant errors peculiar to any observa- tory, when the observation is not made at Greenwich, the ob- served right ascension is to be increased by the average excess for the year (determined by simultaneous obscn^ations) of the right ascensions of the moon's limb made at Greenwich above those made at the actual place of observation. 86a LOXaiTUDE. If now we put m = N = the number of observations = the number of equations of condition, the algebraic sum of the values of ty the sum of the squares of t^ the algebraic sum of the third powers of t^ the sum of the fourth powers of /, the algebraic sum of the values of n, the algebraic sum of the products of n multiplied by t^ the algebraic sum of the products of n multiplied by V^ the normal equations, according to the method of least squares, will be mA+ TB + T^C — J\r = ^ TA+ T^B+ T,C—N,= I (415) T^A+ T^B+ T^C—N^=0 J The solution of these equations by the method of successive substitution, according to the forms given in the Appendix, may be expressed as follows : T.' = T,- m, T,' T,- TT, m t: T,- m r;' — Tl- ( T^y C: rp It -*4 B: n: T' t: c N! = iV, - n; = n,- TN n;'= nj— m T^iV m r.'jv/ t; A = N— T^C — TB m (416) Then, to find the mean error of the corrected Ephemeris, we observe that tliis en'or is simply that of the function .1', which is to be found by the method of the Appendix, acconlin^ to which we first find the coefficients k^y Aj, /*, by the following formulie: vik^ = 1 mt + r; A-, + t;' k. = t* and then, puttmg ^^ - \/W m + A-,« t; + V ^4") BT MOON CULMINATIONS. 861 we have (cX) = Me (417) in which e denotes the mean error of a single observation and (bJC) the mean error of the corrected Ephemeris ; or, if e denotes the probable error of an observation, (eJC) denotes the probable error of the corrected Ephemeris. (Appendix.) If the values of A^, ij, and k^ are substituted in JSf, we shall have It will generally happen, where a suflBcient number of observa- tions are combined, that -^ is a small fraction which may be neglected without sensibly affecting the estimation of a probable error, and we may then take M (418*) According to Peirce, the probable error of a standard observa- tion of the moon's transit is 0M04 (found from the discussion of a large number of Greenwich, Cambridge, Edinburgh, and Wash- ington observations) ; so that the probable error of the corrected Ephemeris will be equal to 31. (0'.104). Example. — At the Washington Observatory, the following right ascensions of the moon were obtained from the transits over twenty-five threads, observed with the electro-chronograph : Approx. Green. Mean Time. R. A. of 3) II Limb. Sid. time semid. passing merid. R. A. of 3) centre 1859, Aug. 16, 19* 0» 8*53'.40 62'.06 0* 7-51*.34 « 17, 20 54 33.57 63.54 53 30.03 " 18, 21 1 42 48.53 65.77 1 41 42.76 The sidereal time of the semidiameter passing the meridian is liere taken from the British Almanac, as we propose to reduce the observations by means of the Greenwich observ^ations which are reduced by this almanac. We thus avoid any error in the semi- diameter. During the semi-lunation from Aug. 13 to Aug. 27, the Greenwich observ^ations, also made with the electro-chronograph, 862 LONGITUDE. gave the following corrections (= n) of the Nautical Almanac right ascensions of the moon : Approx. Greenwich Mean Time. n ( 1859. Aag. 14, 13» — o-.ag — 8. 15, 14 — 0.26 — 1.9 16, 14 — 0.49 — 0.9 18, 16 — 0.63 + 1.2 19, 17 1.04 + 2.2 20, 17 — 1.08 + 3.2 Let us employ these observations to determine by Peirce'a method the most probable correction of the Epliemeris on the dates of the Washington observations. Adopting as the epoch Aug. 17th 12* or 17*'.5, the values of i are approximately as above given. The correction of the Ephemeris being sensibly constant for at least one hour, these values are sufficiently exact. We find then r— 0.8 T, — 29.94 T, = 10.556 7; = 225.045 r;— 29.83 r,' — 6.564 r;= 75.644 Tl'— 74.200 »» = 6 N — — 3'.89 JV, — — 4'.41 N,= — 21'.85 iV;'= — 3.89 N,'- — 2.44 J\r,"— — 1'.58 and hence, by (416), C — 0'.02135 B — — 0: 1257 A = — 0'.525 The correction of the Ephemeris for any given date i^ reckoning from Aug. 17.5, is, therefore, X= — 0'.525 — 0M257f — 0*.02135f» Consequently, for the dates of the Washington ob8er^•ation!», the correction and the probable error (3/£) of the correction, found by (418) or (418*), are as follows: Aug. 16, 19* i = — 0.7 17, 20 t = -\- 0.3 18, 21 t = -\- 1.4 X=-^ 0-.45 X= — 0.56 jr= — 0.74 JlTc = O-.OS Mt = .04 JlTc = .04 The longitude of the Washington Ob8e^^•ato^}• may now lie found by the hourly Ephemeris (after applying these correo- tions), by the method of Art. 234. Taking the obsen-ation of Aug. 16, we have BY MOON CULMINATIONS. 363 Aug. 16, 2; = 19*, R. A. of Ephemeris Ha = 1.8122 da = + 0.0023 a^ log (a, — 0,) ar. CO. log Aa log 60 log of of 1.80774 9.74179 1.77815 3.32768 = 35- 26*.57 = — .80 0* 6- 47*.56 — .45 6 47.11 a^ = 7 51.34 •1 — 0^= 1 4.23 logx'» 6.6554 log^a 7.3617 ar. CO. log Aa 9.7418 log r^Ji 1-1427 log x" n9.9016 x =35 25.77 19» 35- 25'.77 9 37 24.18 3 13.09 Hence, Greenwich mean time = T^-\- x = Sidereal time mean noon Correction for 19* 35- 25*.77 Greenwich sidereal time = Local Bidereal time = 04 = Longitude =5 8 11 .70 5 16 3 .04 7 51.34 The observations of the 17th and 18th being reduced in the same manner, the three results are Probable error.* Weight Aug. 16, 5» 8- ll'.TO 3'.5 1. « 17, 12.50 3.1 1.8 « 18, 11.10 2.9 1.5 Mean by weights = 5 8 11.74 1.8 236. Combination of moon culminations by weights. — When some of the transits either of the moon or of the comparison stars are incomplete, one or more of the threads being lost, such observa- tions should evidently have less weight than complete ones, if we wish to combine them strictly according to the theory of probabilities. Besides, other things being equal, a determina- tion of the longitude will have more or less weight according to the greater or less rapidity of the moon's motion in right ascen- sion. * For the computation of the probable error and weight, see the following article. 864 LONaiTUDE. If the weight of a transit either of the moon or a star were simply proportional to the number of observed threads, as has been assumed by those who have heretofore treated of this sub- ject,* the methods which they have given, and which are obviouB applications of the method of least squares, would be quite suffi- cient. But the subject, strictly considered, is by no means so simple. Let us first consider the formula Oj = a' + dj — ^ or, rather in which t?; and t?' are the observed sidereal times of the transit of the moon and star, respectively ; a' is the tabular right ascen- sion of the star, and a^ is the deduced right ascension of the moon. The probable error of a^ is composed of the probable errors of ??i and of a' — t?', which belong respectively to the moon and the star. We may here disregard the clock errors, as well as the unknown instrumental errors, since tliey aiiect !>| and I?' in the same manner, very nearly, and are sensibly elimi- nated in the difterence &^ — i>'. The probable error of the (luantity a' — i>' is composed of the errors of a' and <?'. Tlie I)robable error of the tabular right ascension of the moon-culmi- nating stars is not only very small, but in the case of correspond- ing observations is wholly eliniinatod ; and even when we use a corrected Ephcmcris it will have but little eftect, since the ol)- sen'^cd right ascension of the moon at the principal ob8er\'atorie8 always depends (or at least should depend) chiefly upon these stars. We may, therefore, consider the error of a' — i>' as sim- plv the error of (?'. We have here to deal with those errors onlv which do not necessarily affect i?' and ??i in the same manner, and of these the chief and onlv ones that need be considered here are — 1st, the culmination error produced by the peculiar con- ditions of the atmosphere at the time of the star's transit, which are constant, or nearly so, during the transit, but are diflferent for different stars and on diftercnt days; and, 2d, the accidental error of observation. It is only the latter which can be diminished * NicoLAi, in the ABtronomitche Xachrichfrn, No. 26; and S. C. WALKsm, tions of the Americ&n Philosophical Society, Vol. VI. p. 258. BT MOON CULMINATIONS. 8G5 by increasing the number of threads. Li Vol. 11. (Transit In- strument) I shall show that the probable error of a single deter- mination of the right ascension of an equatorial star (and this may embrace the moon-culminating stars) at the Greenwich Observatory is O'.OG, whereas, if the culmination error did not exist it would be only 0'.03, the probable error of a single thread being = O'.OS, and the number of threads = 7. Hence, putting c = the probable culmination error for a star^ we deduce* c = l/(0.06)»--(0.03)» = 0'.052 If, then, we put c = the probable accidental error of the transit of a star over a single thread, n = the number of threads on which the star is observed, the probable error of i>', and, consequently, also of a' — t?', is ^ n and the weight of a' — ^' for each star may be found by the formula P= — 7^ in which JE is the probable error of an observation of the weight unity, which is, of course, arbitrary. K we make p = 1 when w = 7, we have E = 0'.06. Substituting this value, and also c = 0'.052, £ = O'.OS, the formula may be reduced to the fol- lowing : P = -^8 (419) 100 + — The value of a^ is to be deduced by adding to ??i the mean * The value of c thus found inyolves other errors besides the culmination error proper, such as unknown irregularities of the clock and transit instrument, &o. These cannot readily be separated Arom c, nor is it necessary for our present purpose. 866 LONGITUDE. according to weights of all the values of a^ — tfj given by the several stars, or where the rectangular brackets are employed to express the sum of all the quantities of the same form. The probable error of the last term will be _ E _ 0v06 "■ VlPl ~ ViP\ K now we put e^ = the probable error of oj, Cj = the culmination error for the moon, he = the probable accidental error of the transit of the moon's limb over a single thread, n^ = the number of threads on which the moon is observed. the probable error of t?i will be = ^/<?i*+ ^ \ and hence i l>] To determine c^ I shall employ the values of the other quantities in this equation which have been found from the Greenwich observations. Professor Peirce gives t^ = 0'.104, and in the cases which I examined I found the mean value k = 1.3. As- suming [p] = 4 as the average number of stars upon which Oj depends in the Greenwich series, we have (0.104). = c,.+ (5:m'+(i:p! whence c, = O'.OOl and the formula for the probable error of a^ observed at the meridian L^ is ,.= („.09,,+ (!^-+C^' (422) In the case of corresponding observations at a second meridian Xj, the probable error e, is also to be found by this formula, and then the probable error of the deduced difference of right ascen- sion will be BY MOON CULMINATIONS. 867 and the probable error of the deduced longitude will be = Ai/e,«+c,« (423) where, If being the increase of the moon's right ascension in 1* of longitude, we have r 3600 ^ = -jy- (424) But if the observation at the meridian ij is compared with a corrected Ephemeris (Art. 235) the probable error of which is J!f (0*.104), the probable error of the deduced longitude will be = h y^fj« + Jf « (0.104)« (425) Finally, all the different values of the longitude will be com- bined by giving them weights reciprocally proportional to the squares of their probable errors. The preponderating influence of the constant error represented by the first term of (422) is such that a very precise evaluation of the other terms is quite unimportant. It is also evident that we shall add very little to the accuracy of an observation by increasing the number of threads of the reticule beyond five or seven. For example, suppose, as in the "Washington observations used in Art. 235, that twenty-five threads are taken, and that four stars are compared with the moon ; we have for each star, by (419), and hence = ^[(„.0„,+ 2f)-H.M.1 = 0..0,, whereas for seven threads we have t^ = 0'.104, and therefore the increase of the number of threads has not diminished the probable error by so much as 0*.01. For the observations of 1859 August 16, 17, 18, Art. 235, the values of A are respectively 82.1 30.8 and 28.8 and, taking Mt = Jff (0'.104) as given in that article, namely, O-.Od 0*.04 and 0'.04 868 LONGITUDE. with the value of e^ = 0*.097 above found, we deduce the proba- ble errors of the tlu'ee values of the lougitude, by (425), 8*.5 3*.l and 2'.9 The reciprocals of the squares of these errors are very nearly in the proportion of the numbers 1, 1.3, 1.5, which were used as the weights in combining the three values. 237. The advantage of employing a corrected Ephemeris instead of corresponding observations can now be determined by the above equations. If the observations are all standard observations (represented by n^=l and [p] = 4), we shall have €i= €3= 0'.104, and the probable error of the longitude will be by corresponding observations = ht^ y/2 by the corrected Ephemeris = he^ y/\ -\- M^ The latter will, therefore, be preferable when JIf < 1, which will always be the case except when very few observations have been taken at the principal observatories. But experience has shown that when we depend wholly on corresponding observations we lose about one-third of the observation.s, and, consequently, the probable error of the final longitude from a series of observations is greater than it would be were all available in the ratio of j 3 : v 2. Hence the proba- ble errors of the final results obtained by corresponding obser>'a- tions exclusively, and by employing the corrected Ephemeris by which all the observations are rendered available, are in the ratio I 3 : ] 1 + J/^, and, the average value of M being about O.G, this is as 1 : 0.G7. If, however, on the date of any given obsen-ation at the meri- dian to be determined, we can find corresponding obserx'ations at two princii)al obsen-atories, the probable error of the longitude found by comparing their mean with the given obser\'ation will be only ht^ ] 1.5, which is so little greater than the avenigo ern>r in the use of the corrected Ephemeris, that it will hardly be worth while to incur the labor attending the latter. If there should l)e three corresponding obsen'ations, the error will be reduced to Asj] 1.33, and, therefore, less than the average error of the corrected Ephemeris. BT MOON CULMINATIONS. 369 The advantage of the new method will, therefore, be felt chiefly in cases where either no corresponding observation, or but one, has been taken at any of the principal observatories. 238. The mean value of h is about = 27, and therefore a probable error of 0*.l in the observed right ascension, supposing the Ephemeris perfect, will produce a mean probable error of 2'.7 in the longitude. K the probable error diminished without limit in proportion to the square root of the number of observa- tions, as is assumed in the theory of least squares, we should only have to accumulate observations to obtain a result of any given degree of accuracy. But all experience proves the fallacy of this law when it is extended to minute errors which must wholly escape the most delicate observation. The remarks of Professor Peirce on this point, in the report above cited, are of the highest importance. He says : " If the law of error embodied in the method of least squares were the sole law to which human error is subject, it would happen that by a sufficient accumulation of observations any imagined degree of accuracy would be attainable in the determination of a constant ; and the evanescent influence of minute increments of error would have the effect of exalting man's power of exact observation to an unlimited extent. I believe that the careful examination of observations reveals another law of error, which is involved in the popular statement that ' man cannot measure what he cannot see.' The small errors which arc beyond the limits of human perception are not distributed according to the mode recognized by the method of least squares, but either with the uniformity which is the ordinary characteristic of matters of chance, or more frequently in some arbitrary form dependent upon individual peculiarities,— such, for instance, as an habitual inclination to the use of certain numbers. On this account, it is in vain to attempt the comparison of the distribution of errors with the law of least squares to too great a degree of minuteness ; and on this account, there is in every species of observation an ultimate limit of accuracy beyond which no mass of accumulated observations can ever penetrate. A wise observer, when he perceives that he is approaching this limit, will apply his powers to improving the methods, rather than to increasing the number of observ^ations. This principle will thus serve to stimulate, and not to paralyze, eflbrt ; and its Vol. L— 24 vivifjing influence will prevent science from stagnating into mere mechanical drudgery. " In approaching the ultimate limit of accuracy, the probable error ceases to diminish proportionably to the increase of the number of observations, eo that the accuracy of the mean of several determinations does not surpass that of the single detCT- minatious as much as it should do in conformity with the law of least squares ; thus it appears that the probable error of the mean of the determinations of the longitude of the Harrard Observatory, deduced from the moon-culminating observation* of 1845, 1846, and 1847, is 1'.28 instead of being I'.OO, to which it should have been reduced conformably to the accuracy of ihe separate determinations of those years. " One of the fundamental principles of the doctrine of probft- bilities is, tliat the probability of an hypothesis is proi>ortionate to its agreement with observation. But any supjtosed computed lunar epoch may be changed by several hundredths of a second without perceptibly attectiug the comparison with observation, provided the comparison is restricted within its legitimate limits of tenths of a second. Ob8er\'ation, therefore, gives no informa- tion which is opposed to such a change." The ultimate limit of accuracy in the deteimination of a longitude by moon culminations, according to the same distin- guished authority, is not less than one second of ii»u. This limit can probably be reached by the observations of t^vo or thre« years, if all the possible ones are taken ; and a longer continaance of them would bo a waste of time and labor. From these considerations it follows that the method of moon culminations, when the transits of the limb are employed, cannot come into competition with the methods by chronometers and occultations where the latter are practicable* * Id oonsequencfl of Ihe uncertainl; atlcnding Iho obMrvtlion et lb* uaa^ of the mgoD'a limb, it Lbs bceo prcipoaed by MaRdLIK [Aitron. AarA. No. SST) to mb- Btituto llio tmnait of a irell-deflDpJ luoar «pot. Tba anlj ftdempi lo carry eat ibU (uggeslion, I ihink, is that of ihe U, S, Coael 8QrT«y. a Kpgri upon vliieb by Mr. I'BTEn* Hill be round in tbe RefioTt of itie Superiatendent for I86n. p. 108. Tba Tsfjiiilg character of a epol as seen in Ip1csoop«s of different powen priaantt. It leems lo me, a *ery formidable obBtule Ut Uie tuccestful appUcation of lUf melliotU BY AZIMUTHS OP THE MOON. 871 FIFTH METHOD. — BY AZIMUTHS OF THE MOON, OR TRANSITS OF THE MOON AND A STAR OVER THE SAME VERTICAL CIRCLE. 289. The travelling observer, pressed for time, will not unfre^ quently find it expedient to mount his transit instniment in the vertical circle of a circumpolar star, without waiting for the meri- dian passage of such a etar. The methods of detennining the local time and the instrumental constants in this case are given in Vol. n. lie may then also observe the transit of the moon and a neighboring star, and hence deduce the right ascension of the moon, which may be used for determining his longitude precisely as the culminations are used in Art. 284. 240. But if the local time is previously determined, we may dispense with all observations except those of the moon and the neighboring star, and then we can repeat the observation several times on the same night by setting the instrument successively in different azimuths on each side of the meridian. It will not be advisable to extend the observations to azimuths of more than 15° on either side. The altitude and azimuth instrument is peculiarly adapted for such observations, as its horizontal circle enables us to set it at any assumed azimuth when the direction of the meridian is approximately known. The zenith telescope will also answer the same purpose. But as the horizontal circle reading is not required further than for setting the instrument, it is not indis- pensable, and therefore the ordinary portable transit instrument may be employed, though it will not be so easy to identify the comparison star. The comparison star should be one of the well-determined moon-culminating stars, as nearly as possible in the same parallel with the moon, and not far distant in right ascension, either preceding or following. The chronometer correction and rate must be determined, with all possible precision, by observations either before or after the moon obsen^ations, or both. An approximate value of the cor- rection should be known before commencing the observations, as it will be expedient to compute the hour angles and zenith distances of the two objects for the several azimuths at which it is proposed to observe, in order to point the instrument properly and thus avoid observing the wrong star. 372 LONGITUDE. To secure the greatest degree of accuracy, the observations should be conducted substantially as follows : — 1st. The instrument being supposed to have a horizontal circle, let the telescope be directed to some terrestrial object, the azimuth of which is known (or to a circumpolar star in the meri- dian), and read the circle. The reading for an object in the meridian will then be known ; denote it by a. 2d. The first assumed azimuth at which the transits are to be observed being Ay set the horizontal circle to the reading ^ + a, and the vertical circle to the computed zenith distance of the moon or tlie star (whichever precedes). This must be done a few minutes before the computed time of the first transit. 3d. Observe the inclination of the horizontal axis with the spirit level. 4th. Observe the transit of the first object over the several threads. 5th. If there is time, observe the inclination of the horizontal axis. 6th. Set the vertical circle for the zenith distance of the second object, and obser\'e its transit. 7th. Observe the inclination of the horizontal axis with the spirit level. The instrument must not be disturbed in azimuth during these operations, which constitute one complete obser\'ation. Now set upon a new azimuth, sufticiently greater to bring the instrument in advance of the preceding object, and repeat the observation. It will often be possible to obtain in this way four or six observations, two or three on each side of the mendian, but the value of the result will not be much increased bv taking more than one obsen*ation on each side of the meridian. The collimation constant is supi)Osed to be known; but<, in order to eliminate any error in it, as well as inequality of pivots, one-half the observations should be taken in each position of the rotation axis. The azimuth of the instrument at each obser\'ation is onlv known from the local time, and hence the following indirei't method of computation will be found more convenient than the usual method of reducing extra-meridian transits; but the reader will find it easy to adapt the methods given in Vol. U. for such purpose to the present case. ^d'- t,f = c,c' = ^f^ = A,A' = BY AZIMUTHS OF THE MOOX. 373 We shall make use of the following notation : Tf T* = the mean of the chronometer times of transit of the moon's limb and the star, respectively, over the several threads,* aT, Ar'= the corresponding chronometer corrections, b,b'= the inclinations of the horizontal axis at the times T and T, c = the collimation constant for the mean of the threads, a, a = the moon's and the star's right ascensions, " " declinations, ** " hour angles, " " true zenith distances, " " parallactic angles, " " azimuths, Aa = the increase of the moon's right ascension in one minute of mean time, a5 = the increase (positive towards the north) of the moon's declination in one minute of mean time, 9r = the moon's equatorial horizontal parallax, S = the moon's geocentric semidiameter, ^ = the observer's latitude, L'= the assumed longitude, ^L=z the required correction of this longitude, L = the true longitude = i' -f ^L, The moon's a, 5, ;r, and S are to be taken from the Ephemeris for the Greenwich time 3^+ a 3^+ ^'(expressed in mean time). The changes Aa, a5 are also to be reduced to this time. The right ascension and declination must be accurately interpolated, from the hourly Ephemeris, with second differences. The quantities j4, ^, y are now to be computed for the chro- nometer time Ty and A\ f ', q' for the time T'. Since A and A' * The chronometer time of passage over the mean of the threads wiU be obtained rigorously by reducing each thread separately to the mean of all by the general formula giren for the purpose in Vol. II. If, however, the same threads are employed for both moon and star, and e denotes the equatorial distance of the mean of the actually obsenred threads from the collimation axis, it will suffice (unless the obserrations are extended greatly beyond the limits recommended in the text) to take the means of the observed times at the times of passage over the fictitious thread the collimation of which is = c. The slight theoretical error which this procedure involves will be eliminated if the observations are arranged symmetrically with respect to the meridian. 374 LONQITUPR are required with all possible precision, logarithms of at least six decimal places are to be employed in their computation ; but for C> ?> C'j ?'> ^^^^ decimal places will suffice. The following formube for this purpose result from a combination of (16) and (20) : For the moon. For the star. tan M= tan (J sec ^ "j . , f tan Jf' = tan d' sec f ^ , tan^cosJbr V ^^tli six I tan ^cos Jf tan -A = -r— -— ( decimals; J tan -A = -:— — - sm (^ — JiT) ) 'I sm (f> — M') (426) tan N = cot ^ cos ^ \ / tan JV^= cot ^ cos f tan ^ sin iV j \ ^ . tan t sin N^ ^°^ = TTTI^f with four ) tan gr' = cos(cJ + i^) decimals- ( co8(^'+^"0 ^ ^ cot (3 + ^ \ '^'''^'°'*^'' i - cot(^-+iVrO tan C = ^ — ' I I tan C = — ^^ — — — • cos q J \ cos ^ in which A and q are to be so taken that sin A and sin q shall have the same sign as sin t. The true azimuth of the moon's limb will be found by applying to the azimuth of the centre the correction S Fupper sign for Ist limbl "" sin C [lower « " 2d " J K we assume the parallax of the limb to be the same as that of the centre (which involves but an insensible error in this case), we next find the apparent azimuth of the limb by applying the correction given by (116), or p7:{<p — <p') sin 1" sin A' cosec C in which f — ^' is the reduction of the latitude, and p is the terrestrial radius for the latitude (p. In this expression we employ A^ which is tlie computed azimuth of the star, for the apparent azimuth of the moon's limb, since by the nature of the obser\'ation they are very nearly equal. To correct strictly for the coUimation and level of the instm- ment, we must have the moon's and star's apparent zenith dis- tances, which will be found with more than sufficient accuracy for the purpose by the fornmlte moon's app. zen. dist. = Ci = C + '^ sm C — refraction star's " " " =C/i=C'— refraction BY AZIMUTHS OF THE MOON. 875 and then the reduction of the true azimuth to the instrumental azimuth (see Vol. IL, Altitude and Azimuth Instrument) is for the mooD; q= sin Ci tan Ci for the star, =p -^ qi sinCi' tanCi' the upper or lower sign being used according as the vertical circle is on the left or the right of the observer. The computed instrumental azimuths are, therefore, . . . , S , fiitCw — e»')8inl"sin^' c b (moon) ill =A ±-:— 1+ ^^ ^f qi -;— -qp 'sinC sine sinCi tanCi ) (427) (star) A,'=A'^^^qz~^ ^ ^ sine/ tanC/ If now the longitude and other elements of the computation are correct, we shall find A^ and A/ to be equal : otherwise, put X = il, — il/ (428) then we are to find how the required correction Ai depends on x, supposing here that all the elements which do not involve the longitude are correct. Now, we have taken a and 8 from the Ephemeris for the Greenwich sidereal time 7" + a T + i', when they should be taken for the time T+ £iT+ i'+ ^L. Hence, if X and ^ denote the increments of the moon's right ascension and declination in one sidereal second, both expressed in seconds of arc, ^ = ^r^ = [9.39675] Aa 60.164 •■ -* we find that '^ = 60^ = ^'-''"^^ ^' a requires the correction X . ^L t « « — ;.aX (429) and these corrections must produce the correction — xm the moon's azimuth. The relations between the corrections of the azimuth, the hour angle, and the declination, where these are so small as to be treated as differentials, is, by (51), 376 LONGITUDE. - . C08^C08flr _ . si dA = dt -\ — Bing dS sin C Bin C that is, cos d cos a , ^ . sin flr ^ _ Sin C sin C Hence, if we put sin C sin C we have aX = - (431) and hence, finally, the true longitude L'+ aX. 241. In order to determine the relative advantages of thia method and that of meridian transits, let us investigate a formola which shall exhibit the effect of every source of error. Let da, 3d, dn, dS = the Corrections of the elements taken from the Ephemeris of the moon, da, dd' = the corrections of the star's place, dT, dT' = the corrections for error in the obs'd time^ d^T =z the correction of A T, dip = the correction of <p. If, when the corrected values of all the elements — ^that of the longitude included— are substituted in the above computation, A^ and -4/ become A^ + dA^ and -4/ + rf-4/, we ought to find, rigorously, A^ + dA, = A^ + dA{ which compared with (428) gives a; = — rf^i + dA^ (432) We have, therefore, to find expressions for dAy^ and dA^' in terms of the above corrections and of aX. We have, first, by diflerentiating (427), sinC siiiC dA^=dA' We neglect errors in c and 6 which are practically eliminated by comparing the moon with a star of nearly the same declina- tion, and combining observations in the reverse positions of the axis. BY AZIMUTHS OF THE MOON. 377 The total difierential of A is, by (51), after reducing dt to arc, dA = -Ibdt '\ dd — cot C sin A dtp sin C sin C consequently, also, .., cos^'cos^f' -.-^ , sing^ ,-, .. . .,, dA = : — —^ . lodf -\ — 7—^ dd' — cot C sin Adtp sm C sin C Since /^=3^4-a3^— a, we have dt = dT+ d^T-'da where rfSTand d^TmsLj be at once exchanged for 5 7" and 3^T; but da is composed of two parts : Ist, the correction of the Ephemeris, and 2d, X (j^L + dT -\- 3£kT)y which results from our having taken a for the uncorrected time. Hence we have, in arc, lbdt = UdT+ 15 ^aT— 15^a — >l(Ai + dT+d/^T) The correction d8 is likewise composed of two parts, namely, dd = dd + ^(Ai + dT+ JaT) Further, we have simply rfJ' = 3d' and df = dT+ ^aT'— ^tt' but, as we may neglect the error in the rate of the chronometer for the brief interval between the observation of the moon and the star, we can take 5a 7'' = 5a T, and, consequently, df=dT'+ d/iT—da' When the substitutions here indicated are made in (432), we obtain the expression x = a/^L + 15f.3a—^^^-^'dd — (15/— a)^r sin C — 15/'.^a'+?^^.o*J'+15/.^T' sin C 07t sin C sin C _ [15 (/_/')- a] 3A r + «i"y-OBin^' g^ (433) sm C sm C 878 LONGITUDE. in which the following abbreviations are used : ^ cos^ cosq cos^'cosj^ sin C Bin C ^ sin flr BinC and in the coefficient of 8(p we have put A = A'. By the aid of this equation we can now trace the effect of each source of error. 1st The coefficients of 55, 55', 5;r, dip have different signs for observations on different sides of the meridian, and therefore the errors of declination, parallax, and latitude will be elimi- nated by taking the mean of a pair of observations equidistant from the meridian. 2d. The star's declination being nearly equal to that of the moon, we shall have very nearly /=/', and the coefficient of 5A7'\\dll be = a; and since to find £^L we have yet to divide the equation by a, it follows that an error in the assumed clock correction produces an equal error (but with a different sign) in the longitude, as in the case of meridian observations. 3d. An error 57^ in the observed time of the moon's transit produces in the longitude the error ( 15/ a I \\dT The mean of the values of a for tvvo observations equidistant from the meridian is Xf. The mean effect of the error 8T \a therefore ( ■*-iUr which is the same as in the ease of a meridian observation. The effect of an error 8T' in the observed time of the star's transit is a and for two observations equidistant from the meridian, the star being in the same parallel as the moon, the mean effect is 'A or X also the same as for a meridian observation. BY AZIMUTHS OF THE MOON. 879 4th. An error 3S in the tabular semidiameter is always elimi- nated in the case of meridian observations when they are com- pared with observations at another meridian, since the same semidiameter is employed in reducing the observations at both meridians. But in the case of an extra-meridian observation the effect upon the longitude b dS dS a sinC ^ cos d cosq — p sinq and in the mean of two observations equidistant from the meridian, the values of q being small, it is (1+2 sin* } q) nearly. X cos d cos q X cos d For a meridian observation the error will be dS Xcoad The error in the case of extra-meridian observations, therefore, remains somewhat greater than in the case of meridian ones, the excess being nearly 2dS.B\n^iq >lC08 d which, however, is practically insignificant ; for we have not to fear that dS can be as great as 1", and therefore, taking q = 15°, 8 = 30°, and >l = 0.4, which are extreme values, the difference cannot amount to O'.l in the longitude. 5th. The ciTor 8a of the tabular right ascension of the moon produces in the longitude the error a and from the mean of two observations equidistant from the meridian, the error is 15 da as in the case of the meridian observation. The error 8a' in the star's right ascension produces the error when the star is in the same parallel as the moon. 880 LONGITUDE. From this discussion it follows that, by arranging the observa- tions symnieirkaUy with respect to the meridian, the mean resuh vn\\ be liable to no sensible errors which do not equally affect meridian observations. But for the large culmination error in the case of the moon (Art. 236), which equally affects extra- meridian observ^ations, the latter would have a great advantage by diminishing the effect of accidental errors. But the probable error of the mean of two obsen^ations equidistant from the meridian, seven threads being employed, will be, by (422), ..= ^[(0.091).+ (2:1^+ ^] = 0..10 and that of a single meridian observation, even where only one star is compared with the moon, is, by the same formula, = O*.!!. When we take into account the extreme simplicity of the computation, the method of moon culminations must evidently be preferred; and that of extra-meridian observations will be resorted to only in the case already referred to (Art. 239), where the traveller may wish to determine his position in the shortest possible time and A\nthout waiting to adjust his instrument accurately in the meridian. Example. — ^At the TJ. S. Naval Academy, 1857 May 9, I ob- serv^ed the following transits of the moon's second limb and of a Scorpiij at an approximate azimuth of 10° East, ^vith an Ertel universal instrument of 15 inches focal length : Chronometer. Lerel. CoUim. D II Limb. T = 16* 11- 30-.17 b = + 2".2 c = 0.0 ) Vertical circle tr Scarpa T = 16 27 49 .83 b' = + 2 .2 ) left. These times are the means of three threads. The chronometer correction, found by tmnsits of stars in the meridian, was — 55"* 9-.16 at 13* sidereal time, and its hourly rate — 0'.32. The assumed latitude and longitude were SP = 38^ 58' 53".5 X' = 5* 5- 55* The star's place was a' = 16* 12- 31'.90 ^' = — 25<^ 14' 58".5 BY AZIMUTHS OF THE MOOX. 881 We first find the sidereal times of the observations of the moon and star respectively, and the Greenwich mean time of the observation of the moon : we have aT = — 65" 9-.89 r + A T = 15* 16~ 20*.28 L'= 5 5 55. Gr. sidereal time = 20 22 15 .28 Sid. time Gr. moon = 3 8 58.91 Sidereal interval = 17 13 16.37 Red. to mean time = — 2 49 .28 Or. mean time = May 9, 17* 10* 27*.09 Hence from the Ephemeris wo find a = 15* 54~ 45'.32 Aa = 2-.1135 S = 14' 47".2 Ar'=— 55~ 9*.97 T' + aT'=15»32-39'.86 d 24*' 42' 54".4 7".619 54' 9".2 By (426) we find A = — log sin q : r sin C Eefraction 9^ 40' 51".0 n9.1581 64^ 19'.5 + 48.8 — 2.1 65 6.2 A' log sin g* Eefraction 9*' 57' 14".8 n9.1719 64^ 54'.1 — 2.1 64 52.0 For the latitude f we find, from Table in., log p = 9.9994 and then, by (427), we find sp— sp'=ll'15" A S = _ 90 40' 5r.o — -r^ = — 16 24 .4 /Mr(f — sin C f ') sin 1" sin A' sin C sinCj 2 .0 .0 1 .0 tan Ci ^= — 9 57 18 .4 il' = — 9^ 57' 14".8 sin C/ .0 1 .0 tan Ci' _____ ^i' = — 9 57 15 .8 B82 LONGITUDE. whence .r = — 2".6 By (429), (430), and (431), we find log ;i = 9.72175 log /9 = n9.10266 a = 0.6064 Ai= — :::^ = --5M4 0.5054 If we wish to see the eftect of all the sources of error in this example, we find, by (433), 0.5054 aL = — 2".6 - 14.96 Aa — 0.16 AA -f- 14.45 AT— 14.82 Sr — 0.36 A^T + 14.82 Aa' + 0.16 AS' + 1.11 AS — 0.001 Air -f 0.002 d# The proper combination of obsci'vations is supposed to eliminate, or at least reduce to a minimum, all the errors except that of the moon's right ascension as given in the Ephemeris. In practice, therefore, it will be necessary to retain the term involving ia. Thus, in the present case we take only 0.5054 Ai = — 2".6 — 14.96 da A second observation on the same day at an azimuth 10** west gave 0.5458 Ai = — 5".7 — 14.92 da The elimination of the errors of declination requires that we take the arithmetical mean of these equations; whence we have, finally, Ai = — 7'.89 — 28.43 ^a SIXTH METHOD. — BY ALTITUDES OF THE MOON. 242. The hour angle (i) of the moon may be computed from an observed altitude, the latitude and declination being known, and hence with the local sidereal time of the obser\'ation (=0; the moon's right ascension by the equation a = — t^ with which the Greenwich time can be found, as in Art 234, and, consequently, also the longitude. The hour angle is most accurately found from an altitude when the observed body is on the prime vertical, and more accurately in low latitudes than in high ones (Art. 149). Tliis method, therefore, is especially suited to low latitudes. The method maybe considered under two forms: — (A) that in which the moon's absolute altitude is directly observed and BT ALTITUDES OF THE MOOX. 883 employed in the computation of the hour angle ; and (B) tliat in which the moon's altitude is compared differentially with that of a neighboring star, — i.e. when the moon and a star are observed either at the same altitude, or at altitudes which differ only by a quantity which can be measured with a micrometer. 243. (A.) Bif the moon's absolute altitude. — This method being practised only with portable instruments, it would be quite superfluous to employ the rigorous processes of correcting for the parallax, which require the azimuth of the moon to be given. The process of Art 97 will, therefore, bo employed in this case with advantage, by which the observed zenith distance is reduced not to the centre of the earth, but to the point of the earth's axis which lies in the vertical line of the observer, and which we briefly designate as the point 0. Let C" = the observed zenith distance, or complement of the observed altitude, of the moon's limb| © = the local sidereal time, L'= the assumed longitude, Ai/ = the required correction of L', L = the true longitude =z L* -\- ^L. Find the Greenwich sidereal time © +-L', and convert it into mean time, for which take from the Ephemeris the quantities 9 = the moon's declination, r = " eq. hor. parallax, S = " semidiameter. Let S' be the apparent semidiameter obtained by adding to S the augmentation computed by (251) or taken from Table XEE. Let r be the refraction for the apparent zenith distance f " ; and put C' = C" + r d= S' (434) Let TTi be the corrected parallax for the point 0, found by (127), or by adding to tt the correction of Table XDI. (which in the present application will never be in error 0".l) ; and put a, = a + e» ffj sin ^ cos J | C, = C'-^,sinC' } (^^^) in which log e*= 7.8244. 384 LONGITUDE. The hour angle (which is the same for the point as for the centre of the earth) is then found by (267), Le. Binie^Jf"°^[^' + (^-''')^""^t-'-(^-^-)J\ (436) \ \ cos f COS ^j / after which the moon's right ascension is found by the formula a = e — < (437) and hence the Greenwich time and the longitude as above stated. But since we have taken 3 for an approximate Greenwich time depending on the assumed longitude, the first computation of i will not be quite correct ; a second one with a corrected value of 3 will give a nearer approximation ; and thus by successive approximations the true value of t and of the longitude will at last be found. But instead of these successive approximations we may obtain at once the coiTCction of the assumed longitude, as follows. We have taken 3 for the Greenwich time © + i', when we should have taken it for the time © + i' + Ai. Hence, putting /9 = the increase of ^ in a unit of time, it follows that o requires the correction ^^L; and therefore, by (51), the correction of the computed hour angle will be cos d tanq in which q is the parallactic angle. Since a = — /, the com- puted right ascension requires the correction (in seconds of time) 15 cos J tanq Therefore, if we put X =z the increase of a in a unit of time, the computed rfreenwioh time and, consequently, also the longi* tude derived from it requires the correction 15 >l cos ^ tang BT ALTITUDES OF THE MOON. 885 Hence, denoting the longitude computed from the right ascen- Bion a = — thj i", we have True longitude = i' + Ai = Z" — fi/^L 15Aco8dtan^ whence Ai = L"—L 1 + /? 15 il sec d cot jT If we denote the denominator of this expression by 1 + a, we shall have, by (18), ^^/tan^^^tan^v 15>l\ sin^ tant / ^ ^ and then ^''"■^' i = i'4.Ai (439) Ai = 1 + a ' Example. — ^At the U. S. Naval Academy, in latitude (p = 88® 58' 58" and assumed longitude i ' = 5* 6"* ()•, I observed the double altitude of the moon's upper limb with a sextant and artificial horizon as below : Chrooometer IlMt LOOiJ BMUlttiM AvnoMd X' Approz. Or. time 1S49 May 2. — Moon east of the meridian. 10* 14» 21*.6 4 41 0.0 6 88 21.6 6 6 0. 10 88 21.6 (For which time we take v, S, and 8 from tlie Nautical Almanac.) a =+ 8«»47'47''.6 «>Viiio ^ oca a » -H 14 .1 a« *- + 8 48 1 .7 Mean of 6 oba. 2 p » 64o 40^ 0^ Indexoorr. ofieztanta — 14 57 App. alt. D Bamn. Alt. Therm. ao**.46 63° F. 660 F. 5— 16'16".4 Air (Tab. xn.)s B'16".4> + 8 .1/ Ajr (Tab. XnL)» + 4 .4 Ivi n » 66 7 .5) S'. sin^'' 2 )64 26 8 . 82 12 81 .6 ' 67 47 28 .6 > + 1 80 i) ' + 16 24 .6 > 58 4 23 .9 47 88 .1 > 57 16 45 .8 With these values of d^, Ci» »»<! 9 = 38° 58' 53", we find, by (436), / = — 3* 19- 53'.64 The sidereal time at Greenwich mean noon, 1849 May 2, was 2*41"'7*.98; whence e= 8»16-14-.61 a = 11 36 8.25 Vol. L— 24 386 LONGITUDE. Corresponding to tliis right ascension we find by the honrij Ephemeris the Greenwich mean time, and hence the longitude i", as follows: Greenwich mean time = 10* 89* 48*.7 Local '* " = 5 33 21.6 X"= 5 6 27.1 L''—L' = + 27M By the hourly Ephemeris we also have for the Greenwich time 10* 39- 48'.7, Increase of a in 1* = ; = -f 2'.014 « ^ in 1- = ^9 = + 10".01 and hence, by (438) and (439), a = — 0.3317 Ai = + 40-.6 i = i' + Ai = 5* 6* 40*.6 244. The result thus obtained involves the errors of the tabular right ascension and declination and the instrumental error. The tabular errors are removed by means of observations of the same data made at some of the principal observatories, as in the case of moon culminations. The instrumental error will be nearly eliminated by determining the local time from a star at the same altitude and as nearly as possible the same declina- tion ; for the instrumental error will then proiiuce the same error in both and /, and, therefore, will be eliminated from their difForcnce — t = a. The error in the longitude will then be no greater than the error in 0. But to give complete effect to this mode of eliminating the error, an instrument, such as the zenith telescope, should be employed, which is capable of indicating the same altitude with great certainty and does not involve the errors of graduation of divided circles. A very ditforcnt method of observation and computation must then be resorted to, which I proceed to consider. 245. (B.) Bi/ equal aliltmlcs of the moon and a star^ observed with (he zauth telescope. — The reticule of this instrument should for these observations be provided with a system of fixed Iiorizontal threads: neveiiheloss, we may dispense with them, and employ only the single movable micrometer thread, by setting it suc- cessively at convenient inten'als. BY ALTITtDES OP THE MOON. 887 Having selected a well determined star as nearly as possible in the moon's path and differing but little in right ascension, a preliminary computation of the approximate time when each body will arrive at some assumed altitude (not less than 10°) must be made, as well as of their approximate azimuths, in order to point the instrument properly. The instrument being pointed for the first object, the level is clamped so that the bubble plays near the middle of the tube, and is then not to be moved between the observation of the moon and the star. After the object enters the field, and before it reaches the first thread, it may be necessary to move the instrument in azimuth in order that the transits over the horizontal threads may all be observed without moving the instrument during these transits. The times by chronometer of the several transits are then noted, and the level is read off. The instrument is then set upon the azimuth of the second object, the observation of which is made in the same manner, and then the level is again read off. This com- pletes one observation. The instrument may then be set for another assumed altitude, and a second observation may be taken in the same manner.* Each observation is then to be separately reduced as follows : Let 1, t', H'y &c. = the distances in arc of the several threads from their mean, m, m' = the mean of the values of i for the observed threads, in the case of the moon and star respectively, /, r= the level readings, in arc, for the moon and star, 0, 0'= the mean of the sidereal times of the observed transits of the moon and star; then the excess of the observed zenith distance of the moon's limb at the time above that of the star at the time ©' isf the quantities m and I being supposed to increase with increasing zenith distance. * The same method of obseryation may be followed with the ordinary universal instrument, but, as the level is generally much smaller than that of the zenith tele- scope, the same degree of accuracy will not be possible. f Wheu the micrometer is set successively upon assumed readings, m and m' will be the means of these readings, converted into arc, with the known value of the screw. 388 LONGITUDE. A](flo, let o, ^, t, C, A, J = the E. A., decl., hour angle, geocentric zenith distance, azimuth, and parallactic angle of the moon's centre at the time ©; o', d', Ify C', A' J g[ = the same for the star at the time O'; 7tj S = the moon's equatorial hor. parallax and semidiameter; X = the increase of a in 1* of sid. time ; fi :=. " ^ « « <« ^ = the latitude ; X' = the assumed longitude; A-L = the required correction of Jj'; The quantities a, d, ;r, and S are to be taken from the Ephemeris for the Greenwich sidereal time + i ' (converted into mean time) ; a and 8 being interpolated with second differences by the hourly Ephemeris. Then the required correction of the longi- tude will be found by comparing the computed value of ^ with the observed value. For this purpose we first compute f and {^' \vith the greatest precision, and also A and q approximately. H the differential formula of the next article is also to be computed, A' and j' will also be required. The most convenient formulae will be — For the moon. For the star. t^e — a f=e'_a' tan M=t&n d sec t ^ -^i. - ( tanalf'= tan S' sec f 8m^co8(c> — M) > , . I < ,, sin^'cosCcp — M') cos C = — ( decimals; ) cosC == - 8in M J K sin M' (440) cos -4= tan (^ — Jf)cotZ \ I cos A' = tan (^ — Jf')cotC' taniV=eot^ cos^ / y^iW^ four ) tan i\r'=: cot ^ cost' tan f sin JV i HopiTnAln- i . # tan f sin ^V tan <7 = \ decimals , j tan ^ = cos {pj^N)) \ cos (^' + X") The zenith distance f thus computed will not strictly correspond to the time unless tlic assumed longitude is correct. Let its true value be ^ + d^. Also put Cj =r the observed zenith distance of the moon's limb, Cj' = the observed zenith distance of the star, r^r' = the refraction for C, and C', BY ALTITUDES OF THE MOON. 889 then Cj =c/+m — m' + Z — r Putting then C"=Ci + r = C +m — m'+/ — r+(r — O and, by Art (136), : (441) r = (^ — 9') 00s A sinjp=/t)8in7r8in(C" — r) k=zpz^ /gq: iijpT^ /S) sin |) sin S the 1 1 \ sign being used for the moon's ^ i \ limb, we have This equation determines rf^. We have, therefore, only to determine the relation between rff and Ai. Now, we have taken a and d for the Greenwich sidereal time @ -\- L'^ when we should have taken them for the time © + i ' + aZ ' : hence a requires the correction k^L a " « i9Ai t " « — AAi> and then, by (51), dZ := — cos q . p^L — sin jr cos ^ . 15 >l ^L Hence, putting x = — rf^, or x = Z — C' + k and a = \blBmq cos ^ + i^ ^^ ? we have i^L = — L = L' + ^L (442) The solution of the problem, upon the supposition that all the data are correct, is completely expressed by the equations (440), (441), and (442). 246. The quantity x is in fact produced not only by the error in the assumed longitude, but also by the errors of observation and of the Ephemeris. In order to obtain a general expression 390 LONGITUDE. in which the eflfect of every source of error may be represented, let T, T'= the chronometer times of observation of the moon and star, A r = the assumed chronometer correction, dTjdT= the corrections of T and T for errors of observation, d^T = the correction of a T, da, dd, dn, dS = the corrections of the elements taken from the EphemeriS; ^^ == the correction of the assumed latitude. K, when the corrected values of all the elements are substituted, ^, f ', k become ^ + rf^, ^' + ^C'* ^ + ^^'> instead of the equation f " —{^ + dC) = k we shall have C"+ (?:' — (C + d:) = k + dk and hence x = — d: + d:'^dk (448) and we have now to find expressions for rf^, df ', and dk in terms of the above corrections of the elements. Taking all the quantities as variables, we have dZ = 15 sin q cos d dt — cos q dd -\- cos A df <fC'= 15 sin 5^ cos ^'rf^ — cos^c/^'-f qo^A* dtp Since ^ = J* + a J* — a, we have dt = dT -\- di^T^da where dT^and rf^T^may be exchanged for 5 7" and 5a J*, but da is composed of two parts: Ist, of the actual correction of the Ephemcris; and 2d, of ^(aZ + dT -{■ d^T) resulting from our having taken a for the uncorrected time : hence we have The correction d8 is also composed of two parts, so that dd = dd + fi(j:iL + dT+ ^aT) Further, we have simply d8^ = Sd\ and df=dT+ dAT^da' in which 5a T at the time T' is assumed to be the same as at the BY ALTITUDES OP THE MOON. 891 time T^ an error in the rate of chronometer being insensible in the brief interval between the observations of the moon and the star. Again, we have, from (441), cos p dp = p cos n sin (C" — z') <i:c + /> sin it cos (C" — r) <^» " dk = dp z^: dS or, with sufficient accuracy, dk = sin C' dn :^ dS -{- %m 'k cos C' dZ' Now, substituting in rff and rfi^' the values of rf/, rf5, &c., and then substituting the values of rff and rff ' thus found, in (443), together with the value of rfA, we obtain the final equation desired, which may be written as follows :* x = a£kL-^f .Ha + co&q.dd —(f^ a)dT — mf.da' — m cos q* dd' + mf . dT — (cos A — m cos A') dip where the following abbreviations are employed : / = 15 sin q cos d f^ =z\h sin g[ cos d' a = Xf -\- fi cosq wi = 1 — sin tt cos C' Having computed the equation in this form, everj' term is to be divided by a, and then aZ/ will be obtained in terms of z and all the corrections of the elements. A discussion of this equation, quite similar to that of (433), will readily show that the observations will give the best result when taken near the prime vertical and in low latitudes, and, farther, that the combination of observations equidistant from the meridian, east and west, eliminates almost wholly errors of declination and parallax and of the chronometer correction. Example.! — At Batavia, on the 11th of October, 1853, Mr. Db Lange, among other observations of the same kind, noted the following times by a sidereal chronometer, when the moon's * The formula (444) is essentially the same as that giyen by Oudemans, Aatronom. Journal, Vol. IV. p. 164. The method itself is the BUggestion of Professor Kaiser of the Netherlands. f Astronomical Journal, Vol. IV. p. 165. 892 LONGirUDE. lower limb and 86 Octpricomi passed the same fixed horizontal i^iireflifls * T = 0* 38- 8'.62 T' = 0» 49- 68'.77 The difierence of the zenith distances mdicated by the level was Z — r = + 2".0 The chronometer correction was aT= + 1"* 8*. 82, and the rate in the interval T' — J* was insensible. The assumed latitude was ^ = — 6® y 57".0 " longitude " i' = -- 7* 7- 37'.0 We have 8 = 0* 39- 11-.94 8'= 0* 50- 67'.09 For the Greenwich sid. time © + i' = 17* 81- 34*.94,or mean time 4* 10- 67*.00, we find, from the Nautical Almanac, a — 21» 12- b'Ab X — + 0'.0387 d — 20° 55' 8".9 i9 — + 0".1440 TT — 57' 51".4 a'= 21* 20- 22'.45 S — 15' 47".8 a'— — 22° 26' 80".5 )utation by (440) gives C — 52° 11' 49".44 C— 53°13'57".30 A — 68° 14'.4 ^'— 66°30'.6 q — 81° 18'.9 g'=80°35'.2 ble TTT. we find ip q>'— 2' 27" log p — 9.999983 Since the same fixed threads were used for both moon and star, we have m = m', and hence also sensibly r = r^; therefore, by (441), we find C" = 53° 13' 59".30 r = — 54".5 p = 46' 21".25 C — C"=— 62' 9".86 A=: + 62'9".17 Hence, by (442), x = ^ 0".69 a = + 0.5575 aZ = — 1'.24 The longitude by this observation, if the Ephemeris is correct, is therefore i =^ Z' + aZ :^ — 7* 7- 38-.24 BY LUNAR DISTANCES. 898 If we compute all the terms of (444), we shall find Ai = — K24 — 24.84 ^a — 0.27 M + 28.84 ^T— 24.24^2" — 0.44 ^aT + 24.28 aa'+ 0.29 ^a'+ 1,79 dS+ IMdn — 0.04^^ This shows clearly the effect of each source of error; but in prac- tice it will usually be sufficient to compute only the coefficients of da and dd. In the present example, therefore, we should take Ai = — 1-.24 — 24.84 da — 0.27 M which will finally be fully determined when da and 88 have been found from nearly corresponding observations at Greenwich or elsewhere. SEVENTH METHOD. — BY LUNAR DISTANCES. 247. The distance of the moon from a star may be employed in the same manner as the right ascension was employed in Arts. 229, &c., to determine the Greenwich time, and hence the longitude. K the star lies directly in the moon's path, the change of distance will be even more rapid than the change of right ascension ; and therefore if the distance could be measured with the same degree of accuracy as the right ascension, it would give a more accurate determination of the Greenwich time. The distance, however, is observed with a sextant, or other re- flecting instrument (see Vol. 11.), which being usually held in the hand is necessarily of small dimensions and relatively infe- rior accuracy. Nevertheless, this method is of the greatest im- portance to the travelling astronomer, and especially to the navigator, as the observation is not only extremely simple and requires no preparation, but may be practised at almost any time when the moon is visible. The Ephemerides, therefore, give the true distance of the centre of the moon from the sun, from the brightest planets, and from nine bright fixed stars, selected in the path of the moon, for every third hour of mean Greenwich time. The planets em- ployed are Saturn, Jupiter, Mars, and Venus. The nine stars, known as lunar-distance stars, are a Arietis, a Tauri {Aldebaran\ P Geminorum (Pollux), a Leonis (Regulus), a Virginis (Spied), a Scorpii (Ajitares), a Aquite (Aliair), a Piscis Australis (Fomal- hau()j and a Pegasi (Markab). The distance observed is that of the moon's bright limb from a Fig. 29. star, from the estimated centre of a planet, or from the limb of the BUn. The apparent distance of the moon's centre from a st«r or planet is found by adding or subtracting the moon's apparent (augmented) semidiameter, according as the bright limb is nearer to or farther from the star or planet than the centre. The ob- served distance of the aun and moon is always that of the nearest limbs, and therefore the apparent distance of the centres is fotuul by adding both seniidiameters.* The apparent distance thus found differs from the (nte (geo- centric) distance, in eonaequeuce of the parallax and refraotioo which affect the altitudes of the objects, and consequently also the distauee. The true distance is therefore to be obtained by computation, the general principle of which may be exhibited tn a simple manner as follows. Let Z, Fig. 29, be the zenith of tlie observer, M' and .S" the ob- served places of the moon and star, S[M' the parallax and refraction of the moon, .*iS' tho refraction of the star, so that M and S are tlie geocentric places. The apparent altitades of the objects nia^v either be measured at the same time as the distance, or, the local time bving known, theymaybe computed (Art. 14). The apparent zenith dis- tances, and, consequently, also the true zenith distances, an? there- fore kno«Ti. Ill the triangle ZM'S' there are known the threo sides, ^/'.Sf' the apparent distance of the objects. 2J/' the apfiarunt zenith distance of the moon, and Z8' the apparent zenith distaoca of the star: from which the angle 2 is computed. Then, in tho triangle ZMS there are known the sides, ZM the moonV true zenith distjnice, and ZS the star's true zenith distance, and the angle Z; from which the required true distance J/.V is coinputcd. In this elementary explanation the parallax and refrnctiou of the moon are supposed to act in the same vertical circle ZM, whereas parallax acts in a circle drawn through the moon and the geocentric zenith (Art, 81 ). while refraction acta in tho vertical circle drawn through the astronomical zenith. Again, when the moon, or the sun, is observed at an altitude less tlmn 50°, it ia necessary to take into account the distortion of tho disc prodaced * We mtij also observe tho tlUlnnee fram the liroh of k plkoat. prorided Ih* tt%- tant lelcsoape is of suHicieai power la gtre ibe pUiiel B vcll-defiDeil dUr ; and iIm pUoet'a scmidiaincter in then also lo b« added or aubtncted. BY LUNAR DISTANCES. 895 by refraction if we wish to compute the true distance to the nearest second of arc (Art. 133). These features, which add very materially to the labor of computation, cannot be over- looked in any complete discussion of the problem. Simple as the problem appears when stated generally, the strict computation of it is by no means brief; and its importance and the frequency of its application at sea, where long computa- tions are not in favor, have led to numerous attempts to abridge it. In most instances the abbreviations have been made at the expense of precision ; but in the methods given below the error in the computation will always be much less than the probable error of the best observation with reflecting instruments : so that these methods are entitled to be considered as practically perfect. With the single exception of that proposed by Bessel,* all the solutions depend upon the two triangles of Fig. 29, and may be divided into two classes, rigorous and approximative. In the rigorous methods the true distance is directly deduced by the rigorous formulae of Spherical Trigonometry ; but in the approxi- mative methods the difference between the apparent and the true distance is deduced either by successive approximations or from a development in series of which the smaller terms are neglected. Practically, the latter may be quite as correct as the former, and, indeed, with the same amount of labor, more correct, since they require the use of less extended tables of logarithms. I propose to give two methods, one from each of these classes. A. — The Rigorous Method, 248. For brevitj', I shall call the body from which the moon's distance is observed tke sun^ for our formulae will be the same for a planet, and for a fixed star they will require no other change than making the parallax and semidiameter of the star zero. * Attron. Nach, Vol. X. No. 218, and Attron. UnUrsuehungfny Vol. II. Bessel's method requires a different form of lunar Ephemeris from that adopted in our Nautical Almanacs. But eren with the Ephemeris arranged as he proposes, the computation is not so brief as the approximative method here giyen, and its supe- riority in respect of precision is so slight as to giye it no important practical advantage. It is, howeyer, the only theoretically exact solution that has been given, and might still come into use if the meaturement of the distance could be rendered much more precise than is now possible with instruments of reflection. 396 LONGITUDE. Let US suppose that at the given local mean time Tihe obser- vation (or, in the ease of the altitudes, computation) has given d"= the apparent distance of the limbs of the moon and sun, h' = the apparent altitude of the moon's centre, H'=z the apparent altitude of the sun's centre, and that in order to compute the refraction accurately the barometer and thermometer have also been observed. For the Greenwich time corresponding to T, which will be found with sufficient accuracy for the purpose by employing the supposed longitude, take from the Ephemeris 8 = the moon's semidiameter, S = the sun's « then, putting d'= the apparent distance of the centres, ^ == the moon's augmented semidiameter, = a + correction of Table XII. we have upper signs for nearest (inner) limbs, lower signs for farthest (outer) limbs. But if the altitude of either body is less than 50°, we must take into account the elliptical figure of the disc produced by refraction. For this purpose we must employ, instead of s' and Sy those seraidiameters which lie in the direction of the lunar distance. Putting q = ZM'S\ Q = ZS'M' (Fig. 29) AS, ^S = the contraction of the vertical semidiameters of the moon and sun for the altitudes h' and H\ the required inclined semidiameters will be (Art 133) s' — AS cos' q and S — aS cos* Q The angles q and Q will be found from the three sides of the triangle ZM'S\ taking for d' its approximate value d" ±: s' ±i S (which is sufficiently exact for this purpose, as great precision in q and Q is not required), and for the other sides 90° — h' and 90° - H\ K we put m = i(A'+ H'+d') BY LUNAR DISTANCES. 397 we shall have . ,- coBm8in(m — IE') . ,,^ cosmsinCm — A') ,..^^ 8mMff = r—^ r; — - Bm«}C= r—jj^ -£^ (445) sin d' cos A' sin d' cos H and then the apparent distance by the formula d' = d" ± (s' — A5 co8« q)±(^8— aS COS* Q) (446) We are now to reduce the distance to the centre of the earth. We shall first reduce it to that point of the earth's axis which lies in the vertical line of the observer. Designating this point as the point 0, Art. 97, let d^, hjj H^ = the distance and altitudes reduced to the point 0, r, R = the refraction for the altitudes h' and H', Xj P z= the equatorial hor. parallax of the moon and sun. The moon's parallax for the point will be found rigorously by (127), but with even more than sufficient precision for the present problem by adding to k the correction given by Table XIII- Denoting this correction by A;r, we have Ai = A'— r + TTi cos (A' — r) J?i = ^' — 72 + Pcos (J' — i?) (447) The parallax P is in all cases so small that its reduction to the point is insignificant. If, then, in Fig. 29, M and S represent the moon's and sun's places reduced to the point 0, and we put Z = the angle at the zenith, MZ8, we shall have given in the triangle M'ZS' the three sides rf', 90° — A', 90° — i?', whence , , - cos } (A' + J' + d') cos } (A' + ^' — d') cos' } z = ^^ ■ ■ > ■ cos A' cos H' and, then, in the triangle MZS we shall have given the angle Z with the sides 90° — Aj and 90° — -Hj, whence the side MS = d^ will be found by the formula [Sph. Trig. (17)], sin* } di = cos' } (Aj + Jli) — cos Ai cos J?i cos' } Z 398 LONOITUDB. To simplify the computation, put m = i(h' + H' + d!) then the last formula, after substituting the value of Z, becomes, sm' J <?i = cos' J (Aj -f J?i) cos m cos (m — <f ) COB A' cos W Let the auxiliary angle M be determined by the equation sin* if = ^Q« ^1 <^Qg -^1 . cos m cos (m — dT) cos A' cos ^' cos' i (Ai + IQ ^ then we have* sin } di = cos J (^ + ^i) cos I£ (449) Finally, to reduce the distance from the point to the centre Fig. 30. of the earth, let P (Fig. 30) be the north pole of the heavens, M^ the moon's place as seen from the point 0, M the moon's geocentric place, S the ^^ sun's place (which is sensibly the same for either point). The point being in the axis of the celestial sphere, the points JIfj and -3f evidently lie in the same declination circle PM^M, Hence, putting d = the geocentric distance of the moon and sun = SM^ d = the moon's geocentric declination = 90° — PAf, ^j =^ the declination reduced to the point = 90® — PJfj, J = the sun's declination = 90° — PS^ we have, in the triangles PMS and M^MS^ ^ ^ cos J. — cos (d, — <5) cos d sin J — sin d cos d cos PMS = ^-^^ = : sin ((Jj — d) sin d cos d sin d We may put cos {d^ — 5) — 1, and, therefore, sin (^, — ^) , . . . , jx cos d^ — cos d = ^^-^ (sm J — am ^ cos a) cos d * This transformation of the formulas is due to Borda, Description et tuagt du ctrtU de rffltxion. BY LUNAR DISTANCES, 399 and since d — d^ is very small, we may put cos d^ — cos d = sin {d — rfj) sin rfj, and hence, very nearly, d-d.^'-!^ (sin J sin d \ sin c?j tan d^ I cos d Substituting the value of ^^ — 3 from (122), ^ J A ' / sin J sin ^ \ ,._^ d — d^ = Ai: Bin ^l ) (450) \ sin d^ tan d^ I in which y is the latitude of the observer, and log A may be taken from the small table given on p. 116. The correction given by this equation being added to d^, we have the geocentric distance d according to the observation. To find the longitude, we have now only to find the Green- wich mean time Tq corresponding to rf, by Art. 66, and then L=T,^T (451) Example. — ^In latitude 35° N. and assumed longitude 150° W., 1856 March 9, at the local mean time jT = 6* 14« 6*, the ob- served altitudes of the lower limbs and the observed distance of the nearest limbs of the moon and sun were as follows, cor- rected for error of the sextant : A" = 52° 34' 0" -ff" = 8° 56' 23" d" = 44° 36' 58".6 The height of the barometer was 29.5 inches, Attached therm. 60° P., External therm. 58° P. I shall put down nearly all the figures of the computation, in order to compare it with that of the approximative method to be given in the next article. Ist. The approximate Greenwich mean time is 5* 14~ 6' + 10* = 15* 14* 6*, with which we take from the American Ephemeris s = 16' 23".l IT = 60' 1".9 ^ = + 14° 19' 8=l& 8".0 P= 8".6 J = — A? 3' 2d. To find the apparent semidiameters, we first take the augmentation of the moon's semidiameter from Table XTT., = 14".0, and hence find «'=16'37'M 400 LONGITUDE. Then to comput<3 the contraction produced by refraction we find from the refraction table, for the given observed altitudes, the contractions of the vertical semidiameters (Art 132), A8 = 0".4 aS = r .6 With the approximate altitudes and distance of the centres we then proceed by (445), as follows : d' = 45° l(y log ooseo d' 0.1498 log cosec d' 0.149S h' =52 51 log sec A' 0.2190 ff' = 9 12 log see JJ' 0.0056 HI =53 37 log COB m 9.7782 log cos m 9.7782 m — ir'=44 26 log sin (m — IT') 9.8460 m — A' = 46 log sin (» — *') aiOS 9.9866 8.0646 log sin }^ 9.9988 log sin } ^ 9.Q27S q= 169«66' Q=: 12? W log cos* (jr 9.9466 log cos* (^ 9.9800 logA« 9.6021 logA5 0.9888 9.6477 a962S Atco8«g= 0."4 A5cos»C= ^-2 Hence we have, by (446), d"=440 86'58".6 a' — AS cos* q =z 16 36 .7 S—aScob'Q= 15 58 .8 ef'=45 9 34.1 3d. To find the apparent and true altitudes of the centres. — ^The apparent altitudes of tlie centres will be found by adding the contracted vertical semidiameters to the ob8er^'ed altitudes of the limbs. The apparent altitudes, however, need not be computed with extreme precision, provided tliat the difl[erenees between them and tlic true altitudes are correct; for it is mainly ui)on these difterenees that the difference between the apparent and true distance depends. The reduction of the moon's horizontal parallax to the point for the latitude 35° is, by Table XIIL, at: = 3".9; and hence we have rj = r + A?: = 60' 5".8 and the computation of the altitudes by (447) is as follows : BT LUNAE DISTANCES. 401 A" = 52*^34' 0" J" = 8° 56' 23" Vert, semid. = 16 37 Vert, semid. = 15 58 h! = 52 50 37 J' = 9 12 21 Table Il.r = 42 .7 E = 5 33 .6 A' — r =52 49 54.3 ff'—B =9 6 47.4 logjT, 3.55700 logP 0.9345 log cos (A' — r) 9.78115 log cos (-ff'— R) 9.9945 3.33815 0.9290 r,cos(A'~r) =__36;j8^5 Pcos(^'— JK) = ^'.5 Ai = 53^ 26' 12".8 H^= 9^ 6' 55".9 4th. We now find the distance d^ by (448) and (449), as follows : d' = 45« 9'34".l A' = 52 50 37 log sec 0.2189683 g'= 9 12 21 log sec 0.0056300 m = 53 36 16 .1 log cos 9.7733154 m — d' = 8 26 42 . log cos 9.9952654 Ai = 53 26 12 .8 log cos 9.7750333 H^= 9 6 55 .9 log cos 9.9944803 2) 9.7626927 9.8813464 i(hi+ -Hi) = 31 16 34 .4 log cos 9.9318007 9.9318007 log sin M 9.9495457 log cos M 9.6583330 idi=22 54 9.2 log sin J (?i 9.5901337 di = 45 48 18 .4 6th. To find the geocentric distance, we have, by (450), for f = 35°, log A 7.8249 d = ^Uo 19/ log IT 8.5565 J=— 4 3 log sin f> 9.7586 1.1400 1.1400 log sin J n8.8490 log sin d 9.3932 log cosec di 0.1445 log cot di 9.9878 n0.1335 n0.5210 — 1".4 — 3".3 ci — ^= — 4".7 d = 45*> 48' 13".7 6th. To find the Greenwich mean time corresponding to rf. Vol. L— 2« 402 LONGITUDE. and hence the lon^tude, according to Art. 66, we find an ap- proximate time {T) + i hj simple interpolation, and then the required time Tq = {T) + t + At, taking At from Table XX., with the arguments t and aQ {= increase of the logarithms in the Ephemeris in 3*), as follows : By the American Ephemeris of 1856 for March 9, we have r) = i5* 0- 0* (d ) = 45^ 40' 54" Q = 0.2510 d =45 48 13 .7 ^ = 13 4 7 19 .7 log — 2.6432 At— 1 log t = 2.8942 To— 15 13 3 T ~ 5 14 6 £ = 9 58 57 B. — The Approximative Method. 249. I shall here give my own method (first published in the Astronomical Journal, Vol. II.), as it yet appears to me to be the shortest and most simple of the approximative methods toheii these are rendered siifficiently accurate by the introduction of all the necessary corrections. Its value must be decided by the im- portance attached to a precise result. There are briefer methods to be found in every work on Navigation, which ^vill (and should) be preferred in eases where only a rude approximation to the longitude is required. As before, let A', H' = the apparent altitudes of the centres of the moon and sun, ei" = the observed distance of the limbs, Sy S = their geocentric semidiameters, r, P = their equatorial horizontal parallaxes, 5' = the moon's semidiameter, augmented by Table XII., TTi =: the moon's parallax, augmented by Table XIII. We shall here also first reduce the distance to the point of Art. 97. The contractions of the semidiamcters prcKluced by refraction will be at first disregarded, and a correction on that account will be subsequently investigated. If then in Fig. 29, p. 394, jl/' and S' denote the apparent places, -Sf and S the places reduced to the point 0, we shall here have BY LUNAR DISTANCES. 408 h! = W^ — ZM\ W = 90«> — ZS\ A,= 90*>— ZM, -Hi = 90°— ZSy and the two triangles give _ cos </, — sin Ai sin H^ cos d* — sin V sin W cos ^ = = ^^^— — — ^— ^— cos hx cos Hx cos A' cos ^' from which, if we put sin A, sin K. cos A, cos K. v(i = * -* n = ^ » sin A' sin -ff' cos A' cos IT we derive cos <i' — COS d^ = (l — n) cos d' + (n — m) sin A' sin H' (a) Put A<f = <fj— d' aA = Aj— A' AH=H'—ff^ (6) then we have cos (i' — cos<fj = 2 sin}A(f sin (4'-f }A<f) (c) and cos(A^+aA) cos (H' — Ag) COS A' cos H' / 2 sin t aA sin (A^+ j aA) \ / 28intAgsin(g^— iAg) \ \ "" cos A' P\ "^ cos J' ) 2 sin t aA sin (A^ + t aA) 2 sin j Ag sin (IT^— t aJ) cos A' cos IT' , 4 sin t A A sin j Affsin (A^ + j a A) sin (H' — } aIT) "' cos A' cos -ET ^^ Also sin A' cos A, sin H' cos jH". — cos A' sin A, cos -EP sin H, n — m = ^ sin A' cos A' sin JET' cos H' Bubstitating in which the values 2 sin A' cos A^ = sin (2 A' -f aA) — sin aA 2 cos A' sin A, = sin (2 A' + aA) -f- sin a A 2sin^'co8^j=sin(2ir'— Al?) + sinA-ff 2 cos JT' sin H^ = sin (2 JT' — aJT) — sin aIT we find sinAg8in(2A^+AA) — sinAA8in(2-EP— aJT) ^"^^ 2 sin A' cos A' sin jff' cos J' ^^^ 404: LONGITUDE. Substituting (c), (rf), and (e) in (a), and at the same time, for brevity, putting . 2 sin i ^A sin (hf + J aA) ^ cos A' ^ _ sin aA sin (2 H' — Ag) * ~" 2 cos A' cos H' ^ _ 2 sin i AJsin (g^— i A^ ' ■" cos H' J. sin aH sin (2 A' + aA) *~~ 2 cos A' cos JET' we have 2 8in}A<fsm(d'+jAcZ)=AiCOsd'+jBi + Cicos(f'+A— ^Cicos^i' CO This formula is rigorously exact ; but, since Arf is always less than 1°, it will not produce an error of O'M to substitute the arcs J Arf, J aA, &c. for their sines, or J Arf sin 1", i aA sin 1", &c. for sin J AC?, sin J aA, &c. ; and therefore we may write /^dHm(d'+i£id)===A^cosd'+Bi + CiCOBd'+I>i^AiCiBinV'co8d' (g) in which A^y B^^ Q, jDp now have the following signification : aA . sin (A' + i aA) cos A -, aA sin(2Jr'— aJT) cos h' 2 cos W A 7T Ci = - ----- . sin (J?'- J AlT) cos if' Aff sin (2 A' + aA) cos H' 2 cos A' The next step in our transformation consists in finding con- venient and at the same time sufficiently accurate expressions of aA and aIT. Let r, R= tho true refractions for the apparent altitades A' and then we have, within less than 0".l, aA = TTi cos (A' — r) — r BY LUNAR DISTANCES. 405 If we neglect r in the term tt^ cos (A' — r), the error in this term will never exceed 1" ; but even this error will be avoided by taking the approximate expression cos (A' — r) = cos A' + sin r sin A' and we shall then have aA = TTi cos A' — r -\- TTi sin r sin A' / If n/i . ^1 ^^^ r 8inA'\ = (ri cos A'— r) 1 H I \ TTi COS A' — r / Since the second term of the second factor produces but 1" in aA, we may employ for it an approximate value, which will still give aA with great precision. Denoting this term by A, we have Tf, sin r sin A' sin r tan A' k = ^1 cos A' — r I ^1 cos A' or, very nearly, /: = sin r tan A' 1 1 -J \ \ TTi cos A' / If we put r = a cot A', in which a has the value given in Table 11., we have A:=asinr(l + — ^\ \ TTi Sin A / Now, a increases with A', but in such a ratio that k remains very nearly constant for a constant value of ;rp We may without sensible error take tt^ = 57' 30" = 3450", which is about the mean value of ;ri, and we shall find for a mean state of the air, by the values of a given in Table IL, forA'= 5° A =z= 0.000291 A' =45 A = 0.000286 A'= 90 k = 0.000285 Hence, if we take k = 0.00029 the formula A A = (tti cos A' — r) (1 + k) (452) will give aA within ^i^Anny ^^ ^^ whole amount, that is, within less than 0".02 in a mean state of the air. For extreme variations 406 LONGITUDB. of the density of the air, it is possible that the refraction may bo increased by its one-sixth part, and k will also be increased by its one-sixth part. But, as the term depending ou k is not more than 1", the error in aA, even in the improbable case supposed, will not be greater than 0'M6. The formula (452) may therefore be regarded as practically exact with the value k = 0.00029. A strict computation of the sun's or a planet's altitude requires the formula A jy = 22 — P cos (J' — R) but P is in all cases so small that the formula £^H = R—Pco^H' (453) will always be correct within a very small fraction of a second. Now, let cos A' cos J?' ' The quantities r' and W computed from the mean values of the refraction are given in Table XTV. under the name "Mean Reduced Refraction for Lunars." The numbers of the table are corrected for the height of the barometer and thermometer by means of Table XIV.A and B. These tables are computed from Bessel*s refraction table, assuming the attached ther- mometer of the barometer, and the external thermometer, to indicate the same temperature, which is allowable in our present problem.* By the introduction of r' and iZ', we obtain £^=(-.-0(1 + A) 7^Jlr--R'-P cos A cos M and the coefficients of formula {g) become * If it is desired to compute r^ and R' with the utmost rigor, it can be done bj Table II., by taking (Art. 107) sin A' sin If' The tables XTV. and XIV.A and B gire the correct Talues to the nearest second in all practical cases. BY LUNAB DISTANCES. 407 Ai = (t:, — r') (1 + A:) sin (A' + i aA) ^ ^ V T y 2 COS J?' Ci = — (i?' — P) sin (J?'— i AlT) Bin(2A-+AA) ^ 2 cos A' The term A^ Q sin 1" cos rf' is very small, its maximnm being only 1". It is easy to obtain an approximate expression for it and to combine it with the term -4iC0srf'. In so small a term we may take Cy sin 1"= — JZ'sin T'sin ^=— sin R tan S'= — A- and hence A, — AlCi sin 1"= ill (1 + A) = (r^ — r') (1 + A)« sin (A' + } aA) If now we put A = (1 + *)' . ?^l^i*l±i^ sin h' sin 2Sr ^_ 8in(g^— JAJT) "" sin^ _ sin (2 h' + aA) "" sin 2 A' ^' = (n^ —r^A sin A' cot d' ^ = — (r, — r') J9 sin J' cosoc d' C"= — {B'—P) C sin jy' cot d' D'= (i?'— P) Dsin A'cosec d' the formula (g) becomes, when divided by sin rf', (455) and (456) sin d' the first member of which may be put under the form ,/, . 2 sin iA</ cos (</'+ iA<f) \ A^l 1 + . ' - \ sm d' I 408 LONGITUDE. 80 that if we put _ AcP sin r^ cos (d' + i^d) ^ "~ 2 Bin d' or, within 0'M5, ar = — J A<f« sin 1" cot d' (457) we have Ad = A'+B'+ C" + D' + X (468) The terms A'y B'^ C", and D' are computed directly from the apparent distance and altitudes by (456), and with sufficient accuracy with four-figure logarithms. The logarithms of ^, jB, C^ D, are given in Table XV., log A and log D with the arguments n^ — r' and A'; log B and log C with the arguments J?' — P and -H'. In the construction of this table a/i and a-BT are com- puted by (452) and (453), and then the logarithms of Aj By Q D, by (455). The sum A'+ B'+C' + D' is called the "first correction of the distance," and, being very nearly equal to Arf, is used as the argu- ment of Table XVI., which gives x, or the " second correction of the distance," computed by (457). When x is greater than 80" and the distance small, it will be necessary to enter this table a second time with the more correct value of Ad found by em- ploying the first value of x. The correction a^ being thus found and added to rf', we have rfp or the distance reduced to the point 0. The reduction to the centre of the earth is then made by (450). This reduction is also facilitated by a table. If we put and then ,_ . / sin J sin d \ N= AttI I \ sin d^ tan <f, / . sin ^ - . sin J a = — A^ = At: tan d^ sin d^ we shall have N=a + b (459) and a and b can be taken from Table XIX. where a is called " the first part of iV^," and b " the second part of N.** We then have d — ^, = iVsin^ (4aO) which is the correction to be added to rfj to obtain the geocentric distance d. Table XIX. is computed with the mean value of BY LUNAR DISTANCES. 409 IT = 57' 30", which will not produce more than 1" error in d — rfj in any case. But, if we wish to compute the correction for the actual parallax, we shall have, after finding N by the table, d-.ef,= iVrflin^X^ (460*) TT being in seconds. The trouble of finding the declinations of the bodies and the HBe of Table XIX. would be saved if the Almanac contained the logarithm of N in connection with the lunar Ephemeris. The value of log iVin the Almanac would, of course, be computed with the actual parallax, and (460) would be perfectly exact. We have yet to introduce corrections for the elliptical figure of the discs of the moon and sun produced by refraction. These corrections are obtained by Tables XVII. and XVJLLl., which aro constructed upon the following principles. Let A5j, aS^= tho contractions of the vertical semidiamoters, ^s, aS = the contractions of the inclined semidiametors; then we have (Art. 133) A5 = A«i cos' q aS= Afif, cos' Q where j = the angle ZM'S' (Fig. 29) and Q = ZS'M'. We have sin H' — sin A' cos d' cos q = cos A' sin d' But, by (456), sin ir J5' sin A' cos rf' A' cos A' sind' B (rj — r') cos A' cos A' sin d' A (r^ — r') cos A' so that cos q = — I 1 ^ \ ^ ^ J9 / (;r^ - r') cos A' If we put A = 1 and jB = 1, which are approximate values, we shall have A'+ B' cos q = -7 ^ (t,— OcosA' ^s = ^s, r ^'+^' T (461) ^L(;rj--r')C08A'J ^ ^ 410 LONGITUDE. In order to ascertain the degree of accuracy of thb formula, we observe that the errors in cos q produced by the assumption A = lj B = ly are .. -.tan A' Bin IT e = (A — l) - — J c' = (1 — jB) tand' cos h* Bind' the errors in cos' q are 2ecosjf 2€^cosq and the errors in a 5 are, therefore, ___ 2as, (A — 1) tan h' cos q , 2as, (1 — ^) sin JT cos ^ * tan d' ^ cos A' sin d' In order to represent extreme cases, let us suppose j = and S^= 90®, which will give e^ and e/ their greatest values; then we shall find for the diflTerent values of A' the following errors : *' e, Un d' e,' sin d' 5» 0".45 0".02 10 .16 .00 15 .08 .00 80 .02 .00 50 .00 .00 it can only be for very small values of rf' that the error e^ can be important, even for A' = 5^ ; and, as these small values of the distance are always avoided in practice, our formula (461) may be considered quite perfect. In the same manner, we shall find » l(^E' — P) cos j&r'J ^ ^ which is even more accurate than (461). These formulae are put into tables as follows. For the moon, Table XVII.A, with the arguments A' and;:^— r', gives the value of 9 = X/ ^ (r^ — O* C08» A' -^ where /is an arbitrary factor (=18000000) employed to give^ convenient integral vahies. Then Table X^TI.B, with the argu- ments^ and A'+ £\ gives BT LUNAR DISTANCES. 411 For the sun, Table XVm. A, with the arguments jff' and JR ' — P, gives the value of AS. in which F= -^; and Table XVm.B gives Cr In these tables A' + B' is called the " whole correction of the moon," and C' + D' the "whole correction of the sun/' As these quantities are furnished by the previous computation of the true distance, the required corrections are taken from the tables without any additional computation. The values of a5 and ^S are applied to the distance as follows : when the limb of the moon nearest to the star or planet is observed, a5 is to be subtracted, and when the farthest limb is observed, a5 is to be added ; when the sun is observed, both ^s and AiS are to be subtracted from d. In strictness, these corrections should be applied to the dis- tance rf', and the distance thus corrected should be employed in computing the values of A\ B\ C", and D'. This would require a repetition of the computation after a5 and ^S had been found by a first computation; but this repetition will rarely change the result by 0".5. In the extreme and improbable case when the distance is only 20° and one body is at the altitude 5® and the other directly above it in the same vertical circle (so that the entire contraction of the vertical semidiameter comes into account), such a repetition would change the result only 1".8 ; and even this error is much less than the probable error of sextant observations at this small altitude, where the sun and moon already cease to present perfectly defined discs. 250. I shall now recapitulate the steps of this method. 1st. The local mean time of the observation being T^ and the assumed longitude i, take from the Ephemeris, for the approxi- 412 LONOrrUDK. mate Greenwich time T+ L^ the quantities 5, Sj ff, P, d, and J. (For the sun we may always take P = 8".5 ; for a star, iS ~ 0, P=0.) 2d. If A'', -ff", d" denote the observed altitudes and distance of the limbs, find s'= 8 -\- correction of Table XIL, 7rj== 7:+ correction of Table XIIL, and the apparent altitudes and distance of the centres, A'=rqis', W=H"^ Sy d'=d"±«'dz S upper signs for upper and nearest limbs, lower signs for lower and farthest limbs. For the altitudes h' and H'^ take the " reduced refractions" r' and R from Table XTV., correcting them by Table XIV.A and B for the barometer and thermometer. Then compute the quantities A' = (tr,— r')^8inA'cotd' C" = — (i?'— P)C8inJrcotrf' B'=^ (r, — r') -B sin W cosec d' D'= (^R'— P)JD sink' cosecd' for which the logarithms of Ay P, Q and D are taken from Table XV. In this table the argument n^ — r' is called the " reduced parallax and refraction of the moon," and P' — Pthe " reduced refraction and parallax of the sun (or planet) or star." For a star this argument is simply P'. When rf'> 90°, the signs of A' and C will be reversed. It may be convenient for the computer to determine the signs by referring to the following table : d' < 90° d' > 90° A' B' C Z)' + — + + + 3d. The terms A* and P', which depend upon the moon's parallax and refraction, may be called the first and second parts of the moon's correction, and the sum A* + P' the " whole cor- rection of the moon." In like manner, C" and P'may be called the first and second parts of the sun's, planet's, or star's correc- BY LUNAR DISTANCES. 418 tioii, and the sura C -|- ly the " whole correction of the sun, planet, or star." The sum of these corrections = ^' + ^' + C + D' may be called the "first correction of the distance.** Taking it as the upper argument in Table XVI., find the second correction = a:, the sign of which is indicated in the table. 4th. Take from Table XVII.A and B the contraction of its inclined semidiameter = a5. If the sun is the other body, take also the contraction from Table A V 111. A and B, = a5. The sign of either of these corrections will be positive when the farthest limb is observed, and negative when the nearest limb is observed. 5th. The correction for the compression of the earth is = J\r sin ifyif being the latitude ; and N may be accurately com- puted by the formula \ sin dy^ tan d^^ I or it may be found within 1" by Table XIX., the mode of con- sulting which is e\ndent. The sign of iVsin ^ will be determined by the signs of iVand sin ^, remembering that for south latitudes sin f is negative. All the corrections being applied to rf', we have the geocen- tric distance d; and hence the corresponding Greenwich time and the longitude. Example. — ^Let us take the example of the preceding article (p. 399), in which the observation gives 1866, March 9th, ^ = 85<>. T = 5* 14" 6* 2 ^" = 52^ 34' 0" Barom. 29.5 in. Assumed i = 10 O -ff"= 8 66 23 Thorm.58^F. Approx. Gr. T. = 15 14 6 B G d" =44 36 58.6 By the Ephemeris, we have s = 16' 23".l r = 60' 1".9 S= 16' 8".0 P= 8".6 Table XII. +14 .0 Tab. XIII. + 3 .9 d = + U^ J= — 4« «'=1637.1 r,= 605.8 The computation may be arranged as follows: 414 LONGITUDE. T)h"= 620 84^,0 fiir''= 80 66'.4 rf"=r44oa6'6r.6 »' = + 16.6 s = 16.1 • = 16 87 .1 A' = 62 60.6 J7' = 9 12.6 5= 16 8.0 d'^45 9 « .7 Table XIV. " " A. ** " B. l'18".l 1 . — 1 . P — 6'49".6 — 6 . 6 . Ill .1 60 6 .8 6 87 .6 8 .6 ^i-r' 68 64 .7 J?' P — 6 29 .0 (Table XV.) log ^ 0.0019 log (ttj — r') 8.6484 log sin h' 9.9016 log cot d' 9.9976 (Table XV.) log C 9.9978 log {If— P) 2.6172 log sin ZT' 9.2042 log cot d' 9.9976 log^' A' — 8.4498 + 46'63".9 logC" fil.7167 52M (Table XV.) log B 9.9981 log (tj — r') 8.6484 log sin W 9.2042 log cosec d' 0.1498 log ^' fi2.9000 -»' = — 13'14".8 -4'+ ^'=-1-83 89 .6 Table XIX. Ist Part of iV= — 6" 2d •* " = — 2 (Table XV.) log Z> 9.9987 log(iZ'— P) 2.6172 log sin h' 9.9016 log ooseo d' 0.1498 log ly 2.6667 lynz-f 6' 8".7 C'+i>'=-f-6 16 .6 Istcorr. = +88'66''J (Table XVI.) 2d corr. = — 18 .5 (Table XVII.) A* = 0. (Table XVIII.) a5= — 9 . 8^0. iV sin ^ = — 4.6 </ = 46 48 12 .8 This result agrees with that found by the rigorous method on p. 401, within 1". To find the longitude, we now have, by the American Ephe- meris for March 9, (T) = 15* 0* 0* (O = 46° 4^ 64" Q = 0.2510 JQ= + n d = 45 48 13 7 19 log = 2.6425 log t = 2.8935 t = 13 3 Table XX. 1 T 15 13 2 T — 5 14 6 L — 9 58 56 BY LUNAR DISTANCES. 415 251. In consequence of the neglect of the fractions of a second in several parts of the above method, it is possible that the computed distance may be in error several seconds, but it is easily seen that the error from this cause will be most sensible in cases where the distance is small ; and, since the lunar distances are given in the Ephemeris for a number of objects, the observer can rarely be obliged to employ a small distance. If he confines himself to distances greater than 46® (as he may readily do), the method will rarely be in error so much as 2", especially if ho also avoids altitudes less than 10*^. Wlien we remember that the least count of the sextant reading is 10'', and that to the probable error of observation we must add the errors of gradua- tion, of eccentricity, and of the index correction, it must be con- ceded that we cannot hope to reduce the probable error of an observed distance below 6", if indeed we can reduce it below 10". Our approximate method is, therefore, for all practical purposes, a perfect method, in relation to our present means of observation. 252. K the altitudes have not been observed, they may be computed from the hour angles and declinations of the bodies, the hour angles being found from the local time and the right ascensions. But the declination and right ascension of the moon will be taken from the Ephemeris for the approximate Green- wich time found wdth the assumed longitude. If, then, the assumed longitude is greatly in error, a repetition of the computation may be necessary, starting from the Greenwich time furnished by the first. As a practical rule, we may be satisfied with the first computation when the error in the assumed longitude is not more than 30*. In the determination of the longitude of a fixed point on land, it will be advisable to omit the observation of the altitudes, as thereby the observer gains time to multiply the observations of the distance. But at sea, where an immediate result is required with the least expenditure of figures, the alti- tudes should be observed. 253. At sea, the observation is noted by a chronometer regu- lated to Greenwich time, and the most direct employment of the resulting Greenwich time will then be to determine the true correction of the chronometer. This proceeding has the advan- 416 LONGITUDE. tage of not requiring an exact determination of the local time at the instant of the observation. For example, suppose the observation in the example above computed had been noted by a Greenwich mean time chrono- meter which gave 15* 10"* ()•, and was supposed to be slow 4" 6*. The true Greenwich time according to the lunar observation was 15* 13"* 0*, and hence the true correction was + S" 0*. With this correction we may at any convenient time afterwards deter- mine the longitude by the chronometer (Art. 214). In this way the navigator may from time to time during a voyage determine the correction of the chronometer, and, by taking the mean of all his results, obtain a very reliable correc- tion to be used when approaching the land. He may even determine the rate of the chronometer with considerable acca- racy by comparing the mean of a number of observations in the first part of the voyage with a similar mean in the latter part of it. 254. To correct the longitude found by a lunar distance for errors of the Ephemeris. — In relation to the degree of accuracy of the observation, we may in the present state of the Ephemeris regard all its errors as insensible except those which affect the moon*8 place. If, therefore, the longitude of a fixed point has been found by a lunar distance on a certain date, the corrections of the moon's right ascension and declination are first to be found for that date from the observations at one or more of the prin- cipal obscrv^atories, and then the correction of the longitude ynXl be found as follows. Let o, o =: the right ascension and declination of the moon given in the Ephemeris for the date of the ohservation, A, il = those of the sun, planet, or star, dafdd=^ the corrections of the moon's right ascension and declination, (id = the corresponding correction of the lunar distance, SL — - the corresponding correction of the computed longi- tude; In Fig. 30, M and S being the geocentric places of the two bodies, as given in the Ephemeris, and d denoting the distance MS, we have cos d = Hin d sin J -f cos (J cos J cos (a — - A) (463) BY LUNAR DISTANCES. 417 by diflferentiating which we find - , cos d cos J sin (a — A) . da = ^^ i . oa sin d COS ^ sin J — sin ^ cos J cos (a — A) .^ ,^^, : — ^^ ^ . M (464) sin a ^ ' If then V = the change of distance in 3*^ we shall have dL = — ddX- (465) in computing which we employ the proportional logarithm of the Ephemeris, Q = log — , reduced to the time of the observation. Example. — At the time of the observation computed in Art. 250, we have Moon, a = 2* 11* 14* ^ = + 14^ 18'.4 Sun, ^ = 23 22 25 j=— 4 3.1 a — ^= 2 49 19 d= 45 48.2 = 42«19'.8 with which we find, by (464), dd = 0.908 ^o + 0.350 M and hence, by (465), with log Q = 0.2511, ^i = — 1.62 ^o — 0.62 dd Suppose then we find from the Greenwich observations da = — O-.SS = — 5".7 and dd = — 4''.0, the correction of the longi- tude above found will be 255. To find the longitude by a lunar distance not given in the Ephemeris. — The regular lunar-distance stars mentioned in Art. 247 are selected nearly in the moon's path, and are therefore in general most favorable for the accurate determination of the Greenwich time. Nevertheless, it may occasionally be found expedient to employ other stars, not too far from the ecliptic. Sometimes, too, a difl'erent star may have been observed by mistake, and it may be important to make use of the observation. Vol. L— 27 418 LONGITUDE. The true distance d is to be found from the observed distance by the preceding methods, as in any other case. Let the local time of the observation be T, and the assumed longitude L. Take from the Ephemeris the moon's right ascension a and de- clination d for the Greenwich time T + L^ and also the star g right ascension A and declination J ; with which the corresp<Mid- ing true distance dgis found by the formula cos d^ = sin ^ sin J -{■ cos d cos A cos (a — A) Then, if d = d^j the assumed longitude is correct ; if otherwise, put ;i = the increase of a in one minute of mean time, fi = the increase of ^ « " *^ " Y = the increase of d " " '* " then we have, by (464), cos d cos J sin (o — A) , cos ^ sin J — sin d cos J cos (a — A) ^ y^^ 1 L,l ; i^ d.jj sin d^ sin d^ and hence the correction of the assumed longitude in seconds of time, r For computation by logarithms, these formulse may be ar- ranged as follows : tan J tanJf = cos (o — A) sin J cos (d — IdTs coBd,= :-^— sm Jf , cos ^ cos J sin (a — -4) , _ ... .. .-.. Bind^ (466) r i Example. — Suppose an observer has measured the distance of the moon from Arciurus, at the local mean time 1856 March 16, r = 10* 30'" 0*, in the assumed longitude i = 6* O* 0*, and, reducing his observ^ation, finds the true distance d = 73« 55' 10" what is the true longitude ? BY LUNAR DISTANCES. 419 For the Greemvich time T+ L = 16^ 30"» we find a = 8* 47- 6-.54 ^ = + 23° 12' 7'M ^ = + 31".40 A= 14 9 7 .04 J = + 19 55 44 .8 /9= — 8 .62 a—A=— 5* 22" 0-.50 = — 80° 30' 7".5 with which we find by (466), d^ = 73° 55' 35". r = — 25".59 d—d^ = — 25" dL = + 58- .6 and therefore the longitude is 6* 0~ 58*.6. 256. In order to eliminate as far as possible any constant errors of the instrument used in measuring the distance, wo should observe distances from stars both east and west of the moon. K the index correction of the sextant is in error, the errors produced in the computed Greenwich time, and conse- quently in the longitude, will have different signs for the two observations, and will be very nearly equal numerically: they will therefore be nearly eliminated in the mean. K, moreover, the distances are nearly equal, the eccentricity of the sextant will have nearly the same effect upon each distance, and will there- fore be eliminated at the same time with the index error. Since even the best sextants are liable to an error of eccentricity of as much as 20", according to the confession of the most skilful makers, and this error is not readily determined, it is important to eliminate it in this manner whenever practicable. K a circle of reflexion is employed which is read off by two opposite verniers, the eccentricity is eliminated from each observation ; but even with such an instrument the same method of observa- tion should be followed, in order to eliminate other constant errors. It has been stated by some writers that by observing distances of stars on opposite sides of the moon we also eliminate a con- stant error of observatiorij such, for example, as arises from a faulty habit of the observer in making the contact of the moon's limb with the star. This, however, is a mistake; for if the habit of the observer is to make the contact too closCy that is, to bring the reflected image of the moon's limb somewhat over the star, the effect will be to increase a distance on one side of the moon while it diminishes that on the opposite side, and the effect upon the deduced Greenwich time will be the same in 420 LONGITUDE. both cases. This will be evident from the following diagram^ (Fig. 31). Suppose a and b ^*^\f^- are the two stars, M the moon's limb. If the observer * [j • judges a contact to exist when the star appears within the moon's disc as at (?, the distance ac is too small and the distance be too great. But, supposing the moon to be moving in the direc- tion from a to 6, each distance will give too early a Greenwich time, for each will give the time when the moon's limb was actually at c. If, however, we observe the sini in both positions, this kind of error, if really constant, will be eliminated ; for, the moon's bright limb being always turned towards the sun, the error will increase both distances, and \vill produce errors of opposite sign in the Greenwich time. Hence, if a series of lunar distances from the sun has been observed, it wnll be advisable to form two distinct means, — one, of all the results obtained from increasing distances, the other, of all those obtained from decreasing dis- tances : the mean of these means ^vill be nearly or quite free from a constant error of observation, and also from constant in- strumental errors. FINDING THE LONGITUDE AT SEA. 257. By chronometers, — This method is now in almost universial use. The form under which it is applied at sea differs verj- slightly from that given in Art. 214. The correction of the chronometer on the time of the first meridian (that of Green- \^'ich among Aniorieau and English navigators) is found at any place whose longitude is known, and at the same time also it<s daily rate is to be established with all possible care. The rate being duly allowed for from day to day during the voyage, the Greenwich time is constantly known, and therefore at anv instant when the local time is obtained bv obser\'ation, the Ion- gitude of the ship is determined. The local time on shipboard is always found from an altitude of some celestial object, observed with the sextant from the sea horizon. (Art. 156.) The computation of the hour angle is then made by (208), and the resulting local time is comjiaretl directly with the Greenwich time given by the chronometer at AT SEA. 421 the instant of the observation. The data from the Ephemeris required in computing the local time are taken for the Green^nch time given by the chronometer. Example. — A ship being about to sail from New York, th<!: master determined the correction on Greenwich time and the rate of his chronometer by observations on two dates, as follows: 1860 April 22, at Greenwich noon, chron. correction = -f S" lO'.O ** " 30, " " " " = + 3 43 .6 Kate in 8 days = + 33 .6 Daily rate = + 4 .2 On May 18 following, about 7* 30* A.M., the ship being in lati- tude 41° 88' N., three altitudes of the sun's lower limb were obsen'ed from the sea horizon as below. The correction of the chronometer on that day is found from the correction on April 30 by adding the rate for 18 days. (It will not usually be worth while to regard the fraction of a day in computing the total rate at sea.) The record of the observation and the whole computa- tion may be arranged as follows : O 290 40/ 10// Barom. 80.32 *». •* 46 Therm. 69« F. «« 60 60 Mean = 29 46 40 Index corr. = — 1 10 Dip = — 4 2 29 40 28 Semid. = + 16 60 Refraction = — 1 42 Parallax = -f- 8 A = 29 64 44 ^ = 41 SS P = 70 21 21 * = 70 64 88 , _ A = 40 69 49 1860 Maj 18. ^ = : 41<» 88' Chronometer 9* 87"« 21'. " 87 63. *• 88 20. Mean — 9 37 61.8 Correction = -f- 4 69.2 Gr. date = Maj 17, 21 42 60.6 for which time we take from the Ephemeris the quantities 0'8 S = 19® 88' 89" Semidiameter = 16' 60" Equation of time = — 3" 49'.8 sec 0.12688 cosec 0.02604 cos 9.61464 •in _9. 81 092 ~9.48348 sin 9.74174 Apparent time = 7*32-* 6».3 Eq. of time = — 3 49.8 Local mean time = 19 28 16.6 Gr. ** " r= 21 42 60.6 Longitude = 2 14 34 = 38« 38'.6 W. In this observation, the sun was near the prime vertical, a posi- tion most favorable to accuracy (Art. 149). 422 LONGITUDE. The method by equal altitudes may also be used for finding the time at sea in low latitudes, as in Arts. 158, 159. 258. In order that the longitude thus found shall be worthy of confidence, the greatest care must be bestowed upon the determination of the rate. As a single chronometer might deviate very greatly without being distrusted by the navigator, it is well to have at least three chronometers, and to take the mean of the longitudes which they severally give in every case. But, whatever care may have been taken in determining the rate on shore, the sea rate will generally be found to differ from it more or less, as the instrument is affected by the motion of the ship ; and, since a cause which accelerates or retards one chro- nometer may produce the same effect upon the others, the agree- ment of even three chronometers is not an absolutely certain proof of their correctness. The sea rate may be found by determining the chronometer correction at t>vo ports whose difference of longitude is well known, although the absolute longitudes of both ports may be somewhat uncertain. For thli pui-pose, a " Table of Chronometric Differences of Longitude*' is given in Raper*s Practice of Navigaiiorij the use of which U illustrated in the following example. Example. — At St. Ilelena, May 2, the correction of a chro- nometer on the local time was — 0* 23'" 10*.3. At the Cape of Good Hope, May 17, the correction on the local time was + 1* 14*^ 28*. 6 ; what was the sea rate ? We have Corr. at St. Ilelena, May 2d = — 0* 23- IC.S Chron. diff. of long, from Raper = + 1 36 45. Corr. for Capo of G. H., May 2d =+ 1 13 34.7 *< " *' " 17th = + 1 14 28 .6 Rate in 15 days = -{- 53 .9 Daily Hoa rate = + 3 .59 259. By lunar distances. — Chronometers, however perfectly made, are liable to derangement, and cannot be implicitly relied upon in a long voyage. The method of lunar distances (Art^ 247-256) is, therefore, employed as an occasional check upon the chronometers even where the latter are used for finding the longitude from day to day. When there is no chronometer on AT 6£A. 423 board, the method of lunar distances is the only regularly avail- able method for finding the longitude at sea, at once sufficiently accurate and sufficiently simple. As a check upon the chronometer, the result of a lunar distance is used as in Art. 253. In long voyages an assiduous observer may determine the sea rates of his chronometers with considerable precision. For this purpose, it is expedient to combine observations taken at various times during a lunation in such a manner as to eliminate as far as possible constant errors of the sextant and of the obser\'er (Art. 256). Suppose distances of the sun are employed exclusively. Let two chronometer corrections be found from two nearly equal distances measured on opposite sides of the sun on two different dates, in the first and second half of the lunation respectively. The mean of these corrections will be the correction for the mean date, very nearly free from constant instrumental and personal errors. Li like manner, any number of pairs of equal, or nearly equal, distances may be combined, and a mean chro- nometer correction determined for a mean date from all the observations of the lunation. The sea rate will be found by comparing two corrections thus determined in two difterent lunations. This method has been successfully applied in voyages between England and India. 260. By the eclipses of Jupiter's satellites, — An observed eclipse of one of Jupiter's satellites furnishes immediately the Green- wich time without any computation (Art. 225.) But the eclipse is not sufficiently instantaneous to give great accuracy ; for, with the ordinary spy-glass with which the eclipse may be observed on board ship, the time of the disappearance of the satellite may precede the true time of total eclipse by even a whole minute. The time of disappearance will also vary with the clearness of the atmosphere. Since, however, the same causes which accele- rate the disappearance will retard the reappearance, if both phenomena are observed on the same evening under nearly the same atmospheric conditions, the mean of the two resulting longitudes will be nearly correct. Still, the method has not the advantage possessed by lunar distances of being almost always available at times suited to the convenience of the na\dgator. 261. By the moon's altitude. — This method, as given in Art. 243, 424 CIRCLES OF POSITION. may be used at sea in low latitudes ; but, on account of tbe unavoidable inaccuracy of an altitude observed from the set horizon, it is even less accurate than the method of the preceding article, and always far inferior to the method of lunar distances, although on shore it is one which admits of a high degree of precision when carried out as in Art. 245. 262. By occultaiions of stars by the moon. — This method, which will be treated of in the chapter on eclipses, may be successfiillj used at sea, as the disappearance of a star behind the moon*t limb may be observed with a common spy-glass at sea with nearly as great a degree of precision as on shore ; but, on account of the length of the preliminary computations as well bb of the subsequent reduction of the observation, it is seldom that a navigator would think of resorting to it as a substitute for the convenient method of lunar distances. CHAPTER VIII. FLJTDING A ship's PLACE AT SEA BY CIRCLES OF POSITION. 263. In the preceding two chapters we have treated of methods of finding the position of a point on the earth's surface by thfi two co-ordinates latitude and longitude; and therefore in all these methods the required position is determined by the inter- section of two circles, one a parallel of latitude and the other a meridian. In the following method it is determined by circles oblique to the parallels of latitude and the meridians. The prin- ciple which underlies the method has often been applied ; but its value as a practical nautical method was first clearly shown by Capt. Thomas II. Sumner.* Let an altitude of the sun (or any other object) be observed at any time, the time being noted by a chronometer regulated to Greenwich time. Suppose that at this Greenwich time the sun ♦ A new and accurate method of finding a thip's position at tea by projection (m tor's chart: by Capt. Thomas H. Sumneb. Boston, 1843. Sumner's method. 425 is vertical to an observer at the point M of the globe (Fig 32). Let a small circle A A' A" be described on the globe from Jif as a pole, with a polar dis- Fig^2. tance MA equal to the zenith distance, or complement of the observed altitude, of the Bun. It is evident that at all places within this circle an observer would at the given time observe a smaller zenith distance, and at all places without this circle a greater zenith distance; and therefore the observa- tion fully determines the observer to be on the circumference of the small circle AA'A'^. If, then, the navigator can project this small circle upon an artificial globe or a chart, the knowledge that he is upon ihi$ circle will bejitst as valuable to him in enabling him to avoid dangers as the knowledge of either his latitude alone or his longitude alone; since one of the latter elements only determines a point to be in a certain circle, without fixing upon any particular point of that circle. The small circle of the globe described from the projection of the celestial object as a pole we shall call a circle of position. 264. To find the place on the globe at which the sun is vertical {or the sun's projection on the globe) at a given Ghreenwich time, — The sun's hour angle from the Greenwich meridian is the Greenwich apparent time. The diurnal motion of the earth brings the sun into the zenith of all the places whose latitude is just equal to the sun's declination. Hence the required projection of the Bun is a place whose longitude (reckoned westward from Green- wich from 0* to 24*) is equal to the Greenwich apparent time, and whose latitude is equal to the sun's declination at that time. 265. From an altitude of the sun taken at a given Greenwich time, to find the circle of position of the observer , by projection on an artificial globe. — ^Find the Greenwich apparent time and the sun's declina- tion, and put down on the globe the sun's projection by the preceding article. From this point as a pole, describe a small circle with a circular radius equal to the true zenith distance deduced from the observation. This will be the required circle of position. 266. Tlie preceding problem may be extended to any celestial 426 CIRCLES OF POSITION. object. The pole of the circle of position will always be th^ place whose west longitude is the Greenwich hour angle of the object (reckoned from 0* to 24*) and whose latitude is the decli- nation of the object. The hour angle is found by Art. 64. 267. To find both the laiiiude and the longitude of a ship by circles of position projected on an artificial globe. — First. Take the altitudes of two different objects at the same time by the Greenwich chronometer. Put down on the globe, by the preceding problem, their two circles of position. The observer, being in the ciream- fcrence of each of these circles, must be at one of their two points of intersection ; which of the two, he can generally determine from an approximate knowledge of his position. Second. Let the same object be observed at two different limefl^ and project a circle of position for each. Their intersection gives the j^osition of the ship as before. If between the observa- tions the ship has moved, the first altitude must be reduced to the second place of observation by applying the correction of Art. 209, formula (380). The projection then gives the ships position at the second observation. 268. From an altitude of a celestial body taken at a given GfreenKick time, to find the circle of position of the observer , by projection on a Mcrcaior chart. — The scale upon which the largest artificial globes are constructed is much smaller than that of the working charts used by navigators. But on the Mercator chart a circle of position will be distorted, and can only Fig. 33. be laid down by points. Let i, i', L" (Fig. 33) be any parallels of latitude crossed by the required circle. For eaoli of these latitudes, with the true altitude found from the observation and the polar distance of the celestial body taken for the Greenwich time, compute the local time, and hence the longitude, '* by chro- nometer'' (Art. 257). Let /, /', /" be the longitudes thus found. Let A, A', A'' be the points whose latitudes and longitudes are, respectively, i, I; L\ V ; L'\ /"; these are evidently points of the required circle. The ship is consequently in the curve AA'A^', traced through these points. . SUMN£R*S METHOD. 427 In practice it is generally sufficient to lay down only twc points ; for, the approximate position of tlie ship being known, if L and U are two latitudes between which the ship may be assumed to be, her position is known to be on the curve AA' somewhere between A and A'. When the difterence between L and U is small, the arc AA' will appear on the chart as a straight line. 269. To find the latitude and longitude of a ship by circles of position projected on a Mercator chart. — IHrst. Let the altitudes of two objects be taken at the same time. Assume two latitudes em- bracing between them the ship's probable position, and find two points of each of their two circles of position by tlie preceding problem, and project these points on the chart. Each pair of points being joined by a straight line, tlie intersection of the two lines is very nearly the ship's position. Thus, if one object gives the points A^ A' (Fig. 34) corresponding to the lati- tudes i, Z/', and the other object the points B^ B' corresponding to the same latitudes, the ship's position is the point C, the intersection of AA' and BB'. It is, of course, not essential that the same latitudes should be used in computing the points of the two circles ; but it is more convenient, and saves some logarithms. If greater accuracy is desired, the circles may be more fully laid down by three or more points of each. Second. — The altitude of the same object may be taken at two different times, and the circles laid down as before ; the usual reduction of the first altitude being applied when the ship changes her position between the observations. It is evident from the nature of the above projection that the most favorable case for the accurate determination of the inter- section C is that in which the circles of position intersect at right angles. Hence the two objects observed, or the two positions of the same object, should, if possible, differ about 90° in azimuth. This agrees with the results of the analytical discussion of the method of finding the latitude by two altitudes. Art. 183. If the chronometer does not give the true Greenwich time, the only effect of the error will be to shift the point C towards the east or the west, without changing its latitude, unless the error U 428 CIRCLES OF POSITION. 80 great as to affect sensibly the deelination which is taken from the Ephemeris for the time given by the chronometer. This method is, therefore, a convenient substitute for the usual method of find- ing the latitude at sea by two altitudes, a projection on the sailing chart being always sufiieient for the purposes of the navigator. Instead of reducing the fii'st altitude for the change of the ship's position between the observations, we may put down the circle of position for each observation and afterwards shift one of them by a quantity due to the ship's run. A' a' ^, Thus, let the first observation give the position line AA' (Fig. 35), and let Aa represent, in direction and length, the ship's course and distance sailed be- tween the observations. Draw aa! parallel to AA'. Then, BB' being the position line by the second observation, its intereection C vnXh aa' is the required position of the ship at the second observation. 270. If the latitude is desired by computation, independently of the projection, it is readily found as follows. Let Zj, Zj = the longitudes (of A and B) found from the first and second altitudes respectively with the latitude X, Zj', Z/ = the longitudes (of A' and B!) found from the same altitudes with the latitude L\ Lq = the latitude of C. From Fig. 34 we have, by the similarity of the triangles ABC mdA'B'C, i;—i;i l^ — l,= B'C : BC whence (^l^^^l^')^(^l^-.l^') s l^ — U=BB' I BC = L'-^L I L^^L Uy^-j^n^^^-^ (467) (^/-O + C^i-'^ (Z'-i)(z,-zo_ h) This formula reduces Sumner's method of " double altitudes** to that given long ago by Lalaxde (Astronomiey Art 3992, and Abrigi de Navigation, p. 68). The distinctive feature of Sumner*s proccfts, however, is that a single altitude taken at any time is made available for determining a line of the globe on which the ship is situated. MERIDUN LINE. 429 271. To find the azimuth of the sun hy a position line projected on the chart. — ^Let AA' (Fig. 36) be a position line on the chart, derived from an observed altitude by Art. 268. At any point C of this line draw CM perpendicular to AA'^ and let NCS be the meri- dian passing through C; then SCM is evidently the sun's azimuth. The line CM is, of course, drawn on that side of the meridian N8 upon which the sun was known to be at the time of the observation. The solution is but approximate, since AA' should be a curve line, and the azimuth of the normal CM" would be diflFerent for different points of AA'. It is, however, quite accurate enough for the purpose of determining the variation of the compass at sea, which is the only practical application of this problem. CHAPTER IX. THE MERroUN LINE AND VARIATION OF THE COMPASS. 272. The meridian line is the intersection of the plane of the meridian with the plane of the horizon. Some of the most use- ful methods of finding the direction of this line will here be briefly treated of; but the full discussion of the subject belongs to geodesy. 273. By th£ meridian passage of a star. — If the precise instant when a star arrives at its greatest altitude could be accurately distmguished, the direction of the star at that instant, referred to the horizon, would give the direction of the meridian line ; but the altitude varies so slowly near the meridian that this method only serves to give a first approximation. 274. By shadoics. — ^A good approximation may be made as follows. Plant a stake upon a level piece of ground, and give it a vertical position by means of a plumb line. Describe one or 430 MERIDIAN line; more concentric circles on the ground from the foot of the stake as a centre. At the tw^o instants before and after noon when the shadow of the stake extends to the same circle, the azimuths of the shadow east and west are equal. The points of the circle at which the shadow terminates at these instants being marked, let the included arc be bisected ; the point of bisection and the centre of the stake then determine the meridian line. Theoretically, a small correction should be made for the sun's change of declina- tion, but it would be quite superfluous in this method. 275. By single altitudes. — ^With an altitude and azimuth instru- ment, observe the altitude of a star at the instant of its passage over the middle vertical thread (at any time), and read the horizontal circle. Correct the observed altitude for refraction. Then, if h = the true altitude, <p = the latitude of the place of observation, jp = the star's polar distance, A = the star's azimuth, A' = the reading of the horizontal circle, we have, from the triangle formed by the zenith, the pole, and the star, tan. M = «i° (^ - f) «'» (^ - ^) (468) cos s cos (s — p) in which s = H9 + ^ + P) In this formula the latitude may be taken with the positive sign, whether north or south, and p is then to be reckoned from the elevated pole ; consequently, also, A will be the azimuth reckoned from the elevated pole. It is evident that in order to bring the telescope into the plane of the meridian we have only to revolve the instrument through the angle A, and therefore either J.' + J. or A' — A, according to the direction of the graduations of the circle, will be the reading of the horizontal circle when the telescope is in the meridian. The same method can be followed when the azimuth is ob- served with a compass and the altitude is measured \vith a sex- tant ; and then A' — A is the variation of the compass. MERIDIAN LINE, 43X 276. From the first equation of (50), ip and 5 being constant, we have (f-4 = — cos A tan ^ and therefore an error in the obsei^ved altitude will have the least eftect upon the computed azimuth when tan y is a maxi- mum; that is, when the star is on the prime vertical. There- fore, in the practice of the preceding method the star should be as far from the meridian as possible. 277. By equal altitudes of a star. — Observe the azimuth of a star with an altitude and azimuth instrument, or a compass, when at the same altitude east and west of the meridian. The mean of the tri'O readings of the instrument is the reading when its sight line is in the direction of the meridian. This is the method of Article 274, rendered accurate by the introduction of proper instruments for observing both the altitude and the azimuth. ft . 278. If equal altitudes of the sun are employed, a correction for the change of the sun's declination is necessary, since equal azimuths will no longer correspond to equal altitudes. Let A' = the cast azimuth at the first observation^ il = " west " " second *^ d = the declination at noon, ^d =: the increase of declination from the first to the second observation, then, by (1), we have, A being the altitude in each case, sin (d — } A^) = sin f %mh — cos ip cos h cos A' sin (d + } A^) = sin ^ sin A — cos ^ cos A cos A the difference of which gives 2 cos 5 sin } A^ = 2 cos <p cos A sin i (u4 + A') sin \{A — A') whence, since a5 is but a few minutes, we have, with sufficient accuracy, A-A'= ^' ""^ ' (469) cos f cos A sin A 482 MEKIDIAN UNS. It will be necessary to note the times of the two observations in order to find a5. If we take half the elapsed time as the hour angle t of the western observation, we shall have, instead of (469), the more convenient formula A-^' = ^^— (470) cos ^ sin t It will not be necessary to know the exact value of k, if only the same bistrumental altitude is employed at both observations. Now let -4/ and A^ be the readings of the horizontal circle at the two observations, then the readings corresponding to equal azimuths are a; and ^^ — (A — A') and, consequently, the reading for the meridian is the mean of these, or That is, the reading for the meridian is the mean of the ob- served readings diminished by one-half the correction (470), We here suppose the graduations to proceed from 0® to 860®, and from left to right. 279. B\j the angular distance of the sun from any terrestrial object. — If the true azimuth of any object in view is known, the direction of the meridian is, of course, known also. The follo^ving method can be carried out with the sextant alone. Measure the angular distance of the sun's limb from any well-defined point of a distant terrestrial object, and note the time by a chronometer. Measure also the angular height of the terrestrial point above the horizontal plane. The correction of the chronometer being known, deduce the local apparent time, or the sun's hour angle t (Art. 54), and then with the sun's declination d and the latitude f compute the true altitude h and azimuth A of the sun by the formuhe (16), or ,^ tan J ^ . tanfcos3f . , .. ,^ j /i-rtx tan Jf= , tan^= -— , tan A ^ cot (f — M)co8A(4il) cos ^ sin (95 — M) Now, let 0, Fig. 37, be the apparent position of the terrestrial point, projected upon the celestial sphere; Sihe apparent place of the sun, Z the zenith, P the pole ; and put MERIDIAN LINE. 433 D = the apparent angular distance of the ^ig. 8^ sun's centre from the terrestrial point = the observed distance increased by the sun's semidiameter, IIz= the apparent altitude of the point, h* = the sun's apparent altitude, a = the difference of the azimuth of the sun and the point, il' = the azimuth of the point. The apparent altitude A' will be deduced from the true altitude by adding the refraction and subtracting the parallax. Then in the triangle 8Z0 we have given the three sides ZS = 90° — A', ZO = 90° — H,80 = D, and hence the angle SZO = a can be found by the formula . ,. sin (5 — -ff) sin (5 — A') ....^ tan' } a = ^^ ^^ ^ (472) cos s cos (s — D) in which Then we have A' = A±a (473) and the proper sign of a to be used in this equation must be determined by the position of the sun with respect to the object at the time of the observation. K the altitude of the sun is observed, we can dispense with the computation of (471), and compute A by the formula (468). The chronometer Avill not then be required, but an approximate knowledge of the local time and the longitude is necessary in order to find d from the Ephemeris. If the terrestrial object is very remote, it will often suffice to regard its altitude as zero, and then we shall find that (472) reduces to tan i a = v^[tan i (D + N) tan J (-Z> — ^0] (474) This method is frequently used in hydrographic surveying to determine the meridian line of the chart. Example. — ^From a certain point 5 in a survey the azimuth of a point C is required from the following observation : Chronometer time =• 4* 12^ 12* Chronom. correction = — 2 Local mean time = 4 10 12 Equation of time = — 4 10 .9 Local app. time, < = 4 6 1 .1 Vol. L— 28 Altitude of C == /]r= 0*> 30* 20" Distance of the nearest limb of the sun from the point C= 48® 17 10" Semidiameter = 16 1 2> = 48 88 11 434 MERIDIAN LINE. The sun's declination was 5=4-4° 16' 55", the latitude was ^ = + 38° 58' 50" ; and hence, by (471), we find A = 74° 36' 36" A = 24° 37' 58" Befraction and parallax = 1 54 K = 2A 39 52 and, by (472), a = 43° 35' 6" Now, the sun was on the right of the object, and hence A'= ^ — a = 31<> 1' 30" Therefore, a line drawn on the chart from B on the left of the line J5C, making with it the angle 81° 1' 30", will represent the meridian. 280. By two measures of the distance of the ^nfrom a terrestrial object. — In the practice of the preceding method with the sextaut, it is not always practicable to measure the apparent altitude of the terrestrial object. We may then measure the distance of the sun from the object at two different times, and, first com- puting the altitude and azimuth of the sun at each observation, we may from these data compute the altitude of the object and the difference between its azimuth and that of the sun at either observation, by formulae entirely analogous to those employed in computing the latitude and time from two altitudes, Art. 178, (304), (305), (306), and (307). 281. By the azimuth of a star at a given time. — ^When the time is known, tlic azimuth of the star is found by (471) : hence we have only to direct the telescope of an altitude and azimuth instrument to the star at any time, and then compare tlie read- ing of its horizontal circle with the computed azimuth. This method will be very accurate if a star near the pole in employed, since in that case an error in the time will produce a comparatively small error in the azimuth. It will be most accu- rate if the star is observed at its greatest elongation, as in the following article. 282. By the greatest elongation of a circumpolar star. — At the instant of the greatest elongation we have, by Art ] 8, cos 8 sin A = cos^ MERIDIAN LINE. . 435 in which A is the azimuth reckoned from the elevated pole. At this instant the star's azimuth reaches its maximum, and for a certain small interval of time appears to be stationary, so that the observer has time to set his instrument accurately upon the star. In order to be prepared for the observation, the time of the elongation must be (at least approximately) known. The hour angle of the star is found by the formula tan w cos t = — tau d and from i and the star's right ascension the local time is founds Art 65. The pole star is preferred, on account of its extremely slow; motion. If the latitude is unknown, the direction of the meridian may nevertheless be obtained by observing the star at both its eastern and its western greatest elongations. The mean of the readings of the horizontal circle at the two observations is the reading for the meridian. 283. One of the most refined methods of determining the direction of the meridian is that by which the transit instrument is adjusted, or by which its deviation from the plane of the meridian is measured ; for which see Vol. IE. 284. At sea, the direction of the meridian, or the variation of the compass, is found with sufficient accuracy by the graphic process of Art 271. ^6 SOLAB BCUPSE8. CHAPTER X. ECUPSES. 285. The term eclipse^ in astronomy, may be applied to any obscuration, total or partial, of the light of one celestial body by another. But the term sohr eclipse is usually confined to an eclipse of the sun by the moon; while an eclipse of the sun by one of the inferior planets is called a transit of the planet. An eclipse of a star or a planet by the moon is called an ocetdiati<m of the star or planet. A lunar eclipse is an eclipse of the moon by the earth. All these phenomena may be computed upon the same genenl principles ; and the investigation of solar eclipses, with which we shall set out, will involve nearly every thing required in the other cases. SOLAS E0LIP8E8. PREDICTION OF SOLAR ECLIPSES FOR THE EARTH GENERALLY. • 286. For the purposes of general prediction, and before enter- ing upon any precise computation, it is convenient to know the limits which determine the possibility of the occurrence of an eclipse for any part of the earth. These limits are determined in the following problem. 287. To find xohether near a given conjunction of the sun and moon^ an eclipse of the sun loill occur. — In order that an eclipse may occur, p. the moon must be near the ecliptic, and, therefore, near one of the nodes of her orbit. Let NS (Fig. 88) be the ecliptic, .V the moon's node, NM the moon's orbit, S and M the centres of the sun and moon at the time of conjunction in longitude, so that MS is a part of a circle of latitude and is perpendicular to GENERAL PREDICTIONS. 487 NS. Let fif' , M', be the centres of the sun and moon when at their least true distance, and put fi z= the moon's latitude at conjunction = SM^ I =z the inclination of the moon's orbit to the ecliptic, ;i = the quotient of the moon's motion in longitude divided by the sun's, Iz= the least true distance = /S'JT, r = the angle SMS\ We may regard NM8 as a plane triangle ; and, drawing M'P perpencticular to NSy we find and hence S'P = |J — 1) tan z' M'P= ^9 — A/9tan r tan Z J«= fi» [(A — 1)« tanV + (1 — A tan Ztan r)'] To find the value of y for which this expression becomes a mini- mum, we put its derivative taken relatively to y equal to zero, whence , X tan I tan y= (il — 1)* + il» tan' J which substituted in the value of 2** reduces it to J» = iS* (^ - 1)» (il — 1)' + A« tan' 7 If then we assume I' such that tanJ'=— ^tanJ (475) we have for the least true distance I = ficoBr (476) The apparent distance of the centres of the sun and moon as seen from the surface of the earth may be less than 2* by the difference of the horizontal parallaxes of the two bodies : so that if we put 7t = the moon's horizontal parallax, Tt' = the sun's " " 438 we have SOLAB ECLIPSBS. minimum apparent distance = 2 — (« — «') An eclipse will occur when this least apparent distance of the centres is less than the sum of the semidiameters of the bodies; and therefore, putting s = the moon's semidiameter, 5'= the sun's " " we shall have, in case of eclipse, or 2: — (tt — ;r')<S + «' fi COSZ' <7r— -7r'4-5 + «' (477) This formula gives the required limit with great precision; but, since i' is small, its cosine does not vary much for different eclipses, and we may in most cases employ its mean value. We have, by observation, Greatest values. Least values. Mean valaes. I 5° 20' 6" 40 57' 22" 5*^ 8' 44" r 61' 32" 52' 50" 57' 11" tt' 9 8 8.5 5 16 46 14 24 15 35 5' 16 18 15 45 16 1 X 16.19 10.89 13.5 From the mean values of / and X we find the mean value of sec /' = 1.00472, and the condition (477) becomes or ^5 < (;r — ;:' + 5 + 5') X 1.00472 /9<7r--7r'+s + s' + (;r — 7r' + 5+«')X .00472 where the small fractional term varies between 20" and 30". Taking its mean value, we have, with sufficient precision for all but very unusual cases, ,5 < r — r' -1- 5 + s' -;- 25" (478) FUNDAMENTAL SQUATXONS. 489 If in this formula we substitute the greatest values of ;r, 5, and s'y and the least value of ;r', the limit /9 < 1<> 34' 53" is the greatest limit of the moon's latitude at the time of cpn- j unction, for which an eclipse can occur. K in (478) we substitute the least values of ;r, 5, and s\ and the greatest value of ;r', the limit /9 < I*' 23' 15" is the least limit of the moon's latitude at the time of conjunc- tion for which an eclipse can fail to occur. Hence a solar eclipse is certain if at new moon ^ < 1° 23' 15", impossible if j9> 1° 34' 63", and doubtful between these limits. For the doubtful cases we must apply (478), or for greater precision (477), using the actual values of ;r, ;r', 5, 5', ^, and / for the date. Example. — On July 18, 1860, the conjunction of the moon and sun in longitude occurs at 2* 19**.2 Greenwich mean time : will an eclipse occur ? We find at this time, from the Ephemeris, /9 = O*' 33' 18".6 which, being within the limit 1° 23' 15", renders an eclipse cer- tain at this time. Having thus found that an eclipse will be visible in some part of the earth, we can proceed to the exact computation of the phenomenon. The method here adopted is a modified form of Bessel's,* which is at once rigorous in theory and simple in practice. For the sake of clearness, I shall develop it in a series of problems. Fundamental Equations of the Theory of Eclipses. 288. To investigate the condition of the beginning or ending of a solar eclipse at a given place on the earth's surface, — The observer sees the limbs of the sun and moon in apparent contact when he is situated in the surface of a cone which envelops and is in contact with the two bodies. "We may have two such cones : * See AttronomUehe Naehrkhtenf Nos. 151, 152, and, for the full development, of the method with the utmost rigor, Bbssbl's Astronomitche Untenuchungeny Vol. II. Hanskn's deyelopment, based upon the same fundamental equations, but theoreti- eally less accurate, may also be consulted with advantage : it is given in Attronom, Ifaeh., Nos. 889-842. 440 SOLAE ECUPSG8. First. The cone whose vertex falls between the sun and the moon, as at V, Fig. 39, and which is called the penumbral cmte. An observer at C, in one of the elements CB V of the cone, see* the points A and B of the limbs of the sun and moon in appaient exterior contact, which is either the first or the last contact ; that is, either the beginning or the ending of the whole eclipse. Fig. SB. Fig. 40. Second. The cone whose vertex is beyond the moon (in the direction of the eartli), as at V, Fig, 40, and which is called the timbral cone, or cone of total shadow. An observer at C in the element CVBA, sees the points A and B of the limbs of the sun and mooii in apparent interior contact, wliich is the beginning or the ending of annular eclipac in case the obser^-er is farther from the moon tlian the vertex of the cone (as in the figure), and whicli is either tlie beginning or the ending of tolcd ccli]>se in case the obseiror is between the vertex of the cone and the moon. If now a plane is passed throngh the point (", at right angles to the axis SVD of the cone, its intersection with the cone will FCNDAMBXTAL EQUATIONB. 441 be a circle (the sun and mooD being regarded ae spherical) whose radius, CD, we shall call the radius of the shadow {peniinibral or umbra]) for that point. The condition of the occurrence of one of the above phases to au observer is, then, that the distance of the point of observation from Ike axis of the shadow is equal to the radius of the shadow for that point. The problems which follow will euable us to traQslate this condition into analytical language. 289. To find for any given time the position of (he axis of the shadow. — The axis of the cone of shadow produced to the celes- tial sphere meets it in that point in which the sun would be projected upon the sphere by an observer at the centre of the moon. Let 0, Fig. 41, be the centre of the earth ; S, that of the sun ; M, that of ^'''; *'■ the moon. The line MS produced to the infinite celestial sphere meets it in the common vanishing point of all lines parallel to MS; that is, in the point Z, in .wliich the line OZ, drawn through the centre of the earth parallel to MS, meets the sphere. The position of the axis of the cone will be determined by tiie right ascension and declination of the point Z. In order to determine the point Z, let the positions of the sun and moon be expressed by rectangular co-ordinates (Art. 32), of which the axis of x is the straiglit line drawn through the centre of the earth and the equinoctial points, the axis of y the inter- flection of the planes of the equator and solstitial colure, and the axis of z the axis of the equator. Let x be taken as positive towards the vernal equinox ; y as positive towards the point of the equator whose right ascension is 90° ; z as positive towards the north. Let a, ^, r ^ the right ascension, declination, and distance fVom the centre of the earth, respectively, of the moon's centre, o',i', r' := the right ascension, declination, and distance fVom the centre of the earth, respectively, of the snn's centre; The co-ordinates x, y, z will be, by (41), 442 SOLAB EOLIPSES. Of the iun. Of the mofm. r' COS d' COS a' r cos d cos a r' COS d' sin a' r cos d sin a r' sin d' r sin ^ Uow let another system of co-ordinates be taken parallel to th« first, the centre of the moon being the origin. The position of the sun in this system will be determined by the right ascension and declination of the sun as seen from the moon ; that is, by the right ascension and declination of the point Z. If we put tty d = tho right ascension and declination of the point Z, G = the distance of the centi*08 of the sun and moon^ the co-ordinates of the sun in the new system are G cos d cos a G cos d sin a G sin d But these co-ordinates are evidently equal respectively to the difference of the corresponding co-ordinates of the sun and moon in the first system ; so that we have G cos d cos a=.i' cos b' coso' — r cos b cosa G cos (i sin a = r' cos b* sin o' — r cos b sin a G ^\Xi d = r' sin 5' — r sin b which fully determine a, rf, and G in terms of quantities which may be derived from the Ephemeris for a given time. But, as a and d differ but little from a' and i', it is exp)edient to put these equations under the following form. (See the similar transformation, Art. 92.) G cos d sin (a — <*') = — ^ ^^^ ^ ^^^ (** — **') G cos d cos (a — o') = r' cos ^' — r cos b cos (a — o') G ^\w d = r' sin ^' — r sin b If these are divided by r', and we put they become g cos d sin {a — a') = — h cos b sin (a — o') g cos d cos (a — a') = cos b* — h cos b cos (o — a') g ^vci d = sin <5' — b sin ^ FUNDAMENTAL EQUATIONS. 443 where the second members, besides the right ascensions and declinations, involve only the quantity 6, which may be expressed in terms of the parallaxes as follows : Let TT =r the moon's equatorial horizontal parallax^ 7r'= the sun's " " " then we have (Art 89) - r sin w' f sin TT K, further, 9ro= the sun's mean horizontal parallax, and r' is expressed in terms of the sun's mean distance from the earth, we have, as in (146), and hence . , sin TT. sm 1^ = — r-^ h = ^^^^ (480) r'siuTT which is the most convenient form for computing 6, because r' and ;: are given in the Ephemeris, and n'o is a constant 290. The equations (479) are rigorously exact, but as 6 is only about jj^, and a — a' at the time of an eclipse cannot exceed 1° 43', a — a' is a small arc never exceeding 17'', which may be found by a brief approximative process with great precision. The quotient of the first equation divided by the second gives . ,, b cos d BCC d' sin (a — a') tan (a — o') = — ^ ^ 1 — b cos d sec d' cos (tt — o') where the denominator diflfers from unity by the small quantity b cos 8 sec d' cos (a — a') ; and, since 8 and i' are nearly equal, this small difterence may bo put equal to 6, and we may then write the formula thus :* a — a' = cos d sec d' (a — o') 1-6 ^ ^ ♦ DeTeloping the formula for tan (a — o') in series, we have , ftcosrfsecfT'sin (a — o') 6' cos' (f sec' d ' sin 2 (a — a') . sin 1" 2 sin 1" whero the second term cannot exceed 0".04, and the third term is altogether inap- 444 SOLAR ECLIPSES. If we take cos {a — a') = 1 and cos (a — a') = 1, we have, from the second and third of (479), g cos d = cos d' — h cos d jT sin d = sin d* — b sin d whence g sin ((Z — ^') = — 6 sin {d — d') g cos (d — ^') = 1 — h cos (d — d') from which follows ^ l_^cos(^ — d') or, nearly,* 1 — 6^ ^ From the above we also have, with sufficient precision for the subsequent application of ^, the formula The formulae which determine the point Z, together with the quantity Gy will, therefore, be a = o' cos d sec d' (o — a') d = 6' ^(c^-O ^ ^""'^ 1 — 6 ^ ^ and in many cases it will suffice to take the extremely simple forms a = a' — 6 (a — a') d = d'^b^d — d') 291. To find the distance of a given place of observation from the axis of the shadow at a given time, — Let the positions of the sun, preciable. The formula adopted in the text is the same as a — a'=— 6co8dsec<5'(a — o') (1 — 6)"* = — b cos 6 sec <5'(o — o') — 6* cos rf sec 6* (a — o') — &o. which, since cos 6 sec i'mny in the second term be put eqnal to unity, dilTers from the complete series only by terms of the third order. The error of the approximate formula is, therefore, something less than (K'.Ol. * The error of this formula, as can be easily shown, will nerer exceed 0^.088. FUNSAUEKTAL EQUATIONS. 445 Ftg. 41 <«»). the moon, and the observer he referred hy rectangular co-ordi- nates to three planes passiug through the centre of tbe earth, of which the plane of xy shall always be at riglit angles to the axis of the shadow, and will here be called the principal plane of refer- awe. Let the plane of yz be the plane of the declination circle passing through the point Z. The plane of xz will, of course, be at right angles to the other two. The axis of z will then be the line OZ, Fig. 41, drawn through the centre of the earth parallel to the axis of the shadow, and will be reckoned as positive towards Z. The axis of y will be the intersection, O Y, of the plane of the declination circle through Z with the principal plane, and will be taken as positive towards the north. The axis of x will he the intersection, OJT, of the plane of the equator with the principal plane, and will he taken as positive towards that point, X, whose right ascension is 90° + a. Let M' and S' he the true places of the moon and sun upon the celestial sphere, P the north pole ; then, if wc put x,y,z ^ the oo-ordinates of the moon, we have, by (Art, 31), X =^r COB M'X y =r COB M' Y z ^ r COB M'Z which, by the formulte of Spherical Trigonometry appUed to the triangles M'PJC, M'PY, M'PZ, become = r COB 3 sin {a — a) = T [sin 9 cos d — cos i si = r [sin i Bin d -|- cos S coa d cos (i B(«-a)] I (48! 8 (a - ay\ ) or x = rcoBi sin (a — a) \ y = r [sin (* — d) cos* i{o — a) -j-sin (3+d)8in'}(o — a)] V (482" z = r [cos(a — d)cos'l(a — a) — co8Ca-i-d)sin'i(o — «)] j and if the equatorial radius of the earth Is taken as the unit of 446 SOLAR ECLIPSES. r, X, y, Zj we shall have the value of r, required in these equa- tions, by the formula r = - — SUITT The co-ordinates x and y of the sun in this system are the same as those of the moon, and the third co-ordinate is z + G; but the method of investigation which we are here following does not require their use. Now let f , ly, C = the co-ordinates of the place of observation, ^ = the latitude of the place, ^z= the reduced latitude (Art. 81), p = the radius of the terrestrial spheroid for the lati- tude f, fi = the given sidereal time ; then, if in Fig. 41 we had taken M for the place of observation, M' would have been the geocentric zenith with the right ascen- sion fi and declination tp'^ and, the distance of the place from the origin being />, we should have found i z= p cos f>' sin (pt — a) '\ 7j = P [sin f ' cos d — cos ^' sin d cos (pt — a)] > (483) C = p [sin ^' sin d -f cos ^ ' cos d cos (pt — a)] j These equations, if we determine A and B by the conditions A sin B = p sin ^ A cos B = p cos ^' cos (pt — a) may be computed under the form S = p cos f ' sin (pi — a) "^ ji = Asin(B — d) >(483*) C = Aco8lB—d) ) The equations (482) might be similarly treated; but the most accurate form for their computation is (482*). The quantity jte — a is the hour angle of the point Z for the meridian of the given place. To facilitate its computation, it is convenient to find first its value for the Greenwich meridian. Thus, if we put for any given Greenwich mean time T fi^ = the hour angle of the point Z at the Greenwich meridian, w = the longitude of the given place. FUNDAMENTAL EQUATIONS. 447 Fig. 42. we have PL— a =A*i"- ^ To find ix^ we have only to convert the Greenwich mean time T into sidereal time and to suhtract a. By means of the formnlse (482) and (483) the co-ordinates of the moon and of the place of observation can be accurately com* puted for any given time. Now, the co-ordinates x and y of the moon are also those of every point of the axis of the shadow : so that if we put A = the distance of the place of observation from the a^s of the shadow, we have, evidently, j«=(x-e)'+(y~^)' (484) [The co-ordinates z and {^ have also been found, as they will be required hereafter.] 292. The distance d may be determined under another form, which we shall hereafter find useful. Let M'j Fig. 42, be the apparent position of the moon's centre in the celestial sphere as seen from the place of observation ; P the north pole ; Z the point where the axis of the cone of shadow meets the sphere, as in Fig. 41 ; Jfj, CJ, the projections of the moon's centre and of the place of observation on the principal plane. The distance C^M^ is equal to J, and is the projection of the line joining the place of observation and the moon's centre. The plane by which this line is projected contains the axis of the cone of shadow, and its intersection with the celestial sphere is, therefore, a great circle which passes through Z, and of which ZM' is a portion. Hence it follows that C^M^ makes the same angle with the axis of 2/ that M^Z makes with PZ: so that if we draw C^N and M^N parallel to the axes of y and z respectively, and put Q = PZM* = NC,M^ we have, from the right triangle C^M^N^ J sin e = a: — f 1 jcose = y-^ I (485) the sum of the squares of which gives again the formula (484). 448 80LAR ECLIPSES. 293. To find the radius of the shadow on the principal plane y or on any given plane parallel to the principal plane, — This radius is evi- dently equal to the distance of the vertex of the cone of shadow from the given plane, multiplied by the tangent of the angle of the cone. In Figs. 39 and 40, p. 440, let EF be the radius of the shadow on the principal plane, CD the radius on a parallel plane draw^i through C Let H = the apparent semidiameter of the sun at its mean dis- tance, k = the ratio of the moon's radius to the earth's equatorial radius, / = the angle of the cone = EVF, c = the distance of the vertex of the cone above the princi- pal plane = FF, C = the distance of the given parallel plane above the prin- cipal plane = DF, I = the radius of the shadow on the principal plane = EF, L = the radius of the shadow on the parallel plane == CD. If the mean distance of the sun firom the earth is taken as unity, we have the earth's radius = sin r^ the moon's radius == k sin r, = MB, the sun's radius = sin J3 = SA, and, remembering that G = r'g found by (481) is the distance MSy we easily deduce from the figures ,i^j.^«-ng±Asin., ^^gg^ r'g in which the upper sign corresponds to the penumbral and the lower to the umbral cone. The numerator of this expression involves only constant quan- tities. According to Bessel, i2'=959".788; Excke found ;r^ = 8".57116 ; and the value of i, found by Burckhardt from eclipses and occultations, is k = 0.27227 ;* whence we have log [sin JE + k sin rj = 7.6688033 for exterior contacts, log [sin H — /: sin ;rj = 7.6666913 for interior contacts. * The ralue of k here adopted is precisely that which the more recent inrestiga- tion of OuDEMANB {Attroti. Naeh., Vol. LI. p. 80) gives for eclipses of the sun. For occultations, a slightly increased value seems to be required. FUNDAMENTAL EQUATIONS. 449. ISoWy taking the earth's equatorial radius as unity, we have sin/ MF = z (Art. 291) and hence c = z±— (487) sin/ ^ ^ the upper sign heing used for the penumbra and the lower for the umbra. We have, then, 1 = c tan/ = z tan/ ± k sec/ L= {c — C) tan/ = I — C tan/ I (488) For the penumbral cone, c — f is always positive, and there- fore L is positive also. For the umbral cone, c — f is negative when the vertex of the cone falls below the plane of the observer, and in this case we have total eclipse : therefore for the case of total eclipse we shall have L = {c — f) tan /a negative quantity. It is usual to regard the radius of the shadow as a positive quantity, and therefore to change its sign for this case ; but the analytical dis- cussion of our equations will be more general if we preserve the negative sign of i as the characteristic of total eclipse. When the vertex of the umbral cone falls above the plane of the observer, L is positive, and we have the case of annular eclipse. For brevity we shall put t = tan/ "I l = ic V (489) L=l^i: ) 294. The analj/iical expression of the condition of beginning or ending of eclipse is or, by (484) and (489), (^ - O' + (y - >?)• =(i- i:y (490) It is convenient, however, to substitute the tvvo equations (485) for this single one, after putting L for J, so that (I — iC) sin Q = X — c (I — iZ) cos Qz=zy — Ti Vol. L— 29 } (491) 450 SOLAK ECLIPSBS. may be taken as the conditions which determine the beginning or ending of an eclipse at a given place. The equation (490), which is only expressed in a different form by (491), is to be regarded as the fundamental equation of the theory of eclipses. 295. By Art. 292, so long as J is regarded as a positive quan- tity, Q is the position angle of the moon's centre at the point Z; and since the arc joining the point Z and the centre of the moon also passes through the centre of the sun, Q is the common position angle of both bodies. Again, since in the case of a contact of the limbs the arc joining the centres passes through the point of contact, Q will also be the position angle of this point when all three points — ^sun's centre, moon's centre, and point of contact — ^lie on the same side of Z. In the case of total eclipse, however, the point of contact and the moon's centre evidently lie on opposite sides of the point Z; and if I — i^ in (490) were a positive quantity, the angle Q which would satisfy these equa- tions would still be the position angle of the moon's centre, but would differ 180° from the position angle of the point of con- tact. But, since we shall preserve the negative sign of / — if for total eclipse (Art. 293), (and thereby give Q values which differ 180° from those which follow from a positive value), the angle Q will in all cases be the position angle of the point of coniacL 296. The quantities a, rf, x, y, /, and i may be computed by the formula (480), (481), (482), (486), (487), (488), for any given time at the first meridian, since they are all independent of the place of observation. In order to facilitate the application of the equations (490) and (491), it is therefore expedient to com- pute these general quantities for several equidistant instauts preceding and following the time of conjunction of the sun and moon, and to arrange them in tables from which their values for any time may be readily found by interpolation. The quantities x and y do not vary uniformly ; and in order to obtain their values with accuracy from the tables for any time, we should employ the second and even the third differences in the interpolation. This is effected in the most simple manner by the following process. Let the times for which x and y have been computed be denoted by T^ — 2*, T^ — 1*, T;, T^ + 1*, FUNDAMENTAL EQUATIONS. 451 T^ + 2*, the interval being one hour of mean time ; and let the values of x and r/ for these times be denoted by a:_2, x^i, &c., y_2, y-i, &c. Let the mean hourly changes of x and y from the epoch T^ to any time T=Tq + rhe denoted by x' and y'. Then the values of x' and y' for the instants T^ — 2*, T^ — 1*, &c. will be formed as in the following scheme, where c denotes the third difference of the values of x as found from the series x_2> ^—h ^^^ according to the form in Art 69, and the difference for the instant T^ is found by the first formula of (77). The form for computing y^ is the same. Time. X *- i;-2» x.t iix,— x_,) T.-l* X-x Xf, — X_i r. X, l(x, — z_,) — \c ^0+1* ^1 x^ — x^ r.+ 2» X, I (x, — x,) If then we require x and y for a time ST = TJ^ + r, we take xf and y' from the table for this time, and we have a: = Xfl 4" ^^ y = yo + y'^ 297. Example. — Compute the elements of the solar eclipse of July 18, 1860. The mean Greenwich time of conjunction of the sun and moon in right ascension is July 18, 2* 8"* 56*. The computation of the elements will therefore be made for the Greenwich hours 0, 1, 2, 8, 4, and 5. For these hours we take the following quantities from the American Ephemeris : M For the Moon. Greenwich meui time. a 6 IT July 18, 0» 116^ 44' 24''.30 21° 52' 20".3 59* 45".80 1 117 21 59 .10 42 82 .8 47 .13 2 117 69 30 .46 82 86 .4 48 .44 8 118 86 58 .85 22 31 .2 49 .72 4 119 14 22 .65 12 17 .2 50 .98 5 119 51 48 .85 1 54 .6 62 .22 »• ^'iu.«^ is"^ :Pr==r. fT Il#^* «L imtt — c ^ -- iu-j > V ^ ^ ^^^^1 ^ i!K -I'.f^ JLi*= ~ w-' JO J- JUvBHlt ■» ILf * 1: ^> ■ji ll!^ -za. -fl \ -t «i :^ rr i « -^ I -I-f ' 14 r- in ij TTj £ ^ 11-? > « -» >i ? .^ :^ Ilf li :^ .Ki 54 41 J4 •IG , -^ ',* i*~" T. dfr- •' '•*■ T, :: Ltl^^I 'V\^ Oft iLcT^ arai. ?:c f'*i& r '/ 1 // i:»r an Tj = 5.»51*94 — »* cr. aearlTT. i = »' — 5 » — i = ?' — 6i-» — r, // I // y I r n't ft (d d) t'Jm^ i(a r cJfH (d d) iihH^ i(a a) -;- r sin (^ — J) sin' } (o — a) aj — r cos (^ — d) sin' \{a — a) III. TJm? an^I<j of the cone of shadow and the radius^f the for pmuml/ra: or exterior rontacU. [7.(i0HS0«] hiu f : For umbra: or mierior contaeti sin . __ [7.666691] C'^- z -\ , log k = 9.435000, c = z Hin / sin/ I = tun/ i= ic i == tan f l = %c FUNDAMENTAL EQUATIONS. 453 IV. The values of a, rf, x, y, log i, and I, will then be tabulated and the differences x' and y' formed according to Art. 296. I give the computation for the three hours 1*, 2*, and 3*, in extenso. I. Elements of ike point Z. (2) a — a <J— (J' log ooseo TT = log r ar. CO. log r' Constant log sin tt^ log* ar. CO. log (1 — b log cos log sec (V log (a — o') log (a — tt') a — a' 'I (1) + (2) log (.5 log(rf d a d log (1 — 6) = log g 1* 2» 0® 40^ 13". 40 -^ 6'12".69 -f 46 3 .38 -f- 85 33 .80 1.7696999 1.7696414 9.9930339 9.9930353 6.61894 7.87167 7.37152 0.001028 0.001028 9.96806 9.96856 0.02978 0.02970 nd. 38263 n2.49511 0.76310 9.86690 4- 5". 66 4- 0".78 7.37269 7.37254 3.43191 8.32915 n0.80460 n0.70169 6". 88 — 6".08 118° 2' 18". 16 118° 4'48".87 20 67 28 .04 20 66 67 .67 9.998977 9.998977 8* +0° 29' 44".58 + 25 65 .45 1.7593866 9.9980367 7.37186 0.001022 9.96905 0.02968 8.26154 nO.62266 — 4".19 7.87238 8.19185 nO. 56428 — 8".67 118° r 9".68 20 66 82 .08 9.998978 n. Co-ordinates x, y, and z. a — a 6--d log sin (tt — a\ log cos o log r COS 6 sin (a — a) = log x X logcos'^ftt — a) log 8in(d — d log(8)=logr8in(<J— </)co8«}ra— a log sin' J (a — a log sin (6 -\- d) log (4) =log r sin (<J -f d) sin* }(*—«) (8) + (4) (3) (4) log cos [6 — d\ log (5)= log r cos (d — rf)co8'J(o — a) log cos (c5 -f ^) log (6) =logr cos (6 -f- rf)8in* \ (a — a) log [(6) - (6)] = log z — 0° 40* 19".06 + 45 9 .76 42 39 66 .84 n8.0692116 9.9680502 n9. 796961 7 —0.626559 9.9999850 8.1184932 9.8781781 6.5363780 9.8310485 7.1271264 -f 0.755402 -1-0.001840 -hO.756742 9.9999626 1.7596474 9.8664780 7.1625559 1.7596364 — 0® 5'18".42 4- 85 38 .88 42 29 33 .97 n7.1817014 9.9685481 n8. 9097909 —0.081244 9.9999998 8.0167484 9.7762846 8.7613394 9.8296236 6.3405048 -j-0. 696058 4-0.000022 4-0.696075 9.9999766 1.7596178 9.8676822 5.8885630 1.7695176 4-0° 29* 48". 77 4- 25 59 .12 42 19 8 .28 7.9381289 9.9690490 9.6665594 0.464044 9.9999920 7.8784602 9.6878287 6.2741910 9.8281695 6.8617470 4-0.434829 4-0.000727 4-0.486056 9.9999876 1.7698661 9.8688989 6.9024714 1.7503601 454 SOLAR SCLIPSBS. HI. Log i and Z, for exterior contacts. [Constant log = 7.668803] logr*^ Const. log r'sf — log Bin/ log Bee/ log k coseo / log [2 + * coBec/] = logc log tai ^/ = log t log ie -log/ / 1* 2» » 0.005943 0.005942 0.005941 7.662860 7.662861 7.66280 0.000006 1.772140 1.772189 1.772188 2.066963 2.066904 2.066826 7.662865 7.662866 7.662867 9.729828 9.729770 9.72969S 0.586819 0.586747 a686652 Log i and I for interior contacts. [Constant log = 7.666691] Const. — log r*^ = log sin/ log sec/ log k cosec / log [« — k cosec/] = log e log tan/ = log t log te == log / / 7.660748 0.000005 1.774252 nO.298985 7.660758 n7. 964788 —0.009010 7.660749 1.774261 n0.297418 7.660754 n7.958167 —0.009082 7.660750 1.774250 fiO.801919 7.660766 fi7.962674 —0.009176 IV. The computation being made for the other hours in the same manner, the results are collected in the following tables. 0* 1 2 3 4 6 a d Exterior Oontacta. Interior CooUcta. 1 I log< I lOf » 117°69'62".44 118 2 18 .16 4 43 .87 7 9.58 9 85 .27 12 .95 20° 57' 48".60 67 23 .04 56 57 .57 56 82 .08 56 6 .68 66 41 .06 0.686867 0.586819 0.636747 0.586652 0.586583 0.686891 7.662864 65 66 67 68 69 — 0.008960 0.009010 0.009082 0.009176 0.009298 0.009434 7.660752 58 64 , 55 56 57 ; • 0* 1 2 8 4 5 X A. A. — 27 — 87 162 A. —46 —60 —76 y A. A. 1 A. 1.171856 0.626659 — 0.081244 -1- 0.464044 4- 1.009245 -h 1.664284 4- 0.646297 0.646816 0.646288 0.646201 0.546089 -f- 0.917040 l-f- 0.756742 l-f 0.596076 -1-0.485066 -j- 0.278704 !-f 0.1 12089 — 0.160298 0.160667 — 0.161019 — 0.161852 — 0.161666 369 862 —888 —818 -K -19' For the values of the hourly differences of x and y, we find from the above, by Art. 296, FCyDAMENTAL EQUATIONS. 455 i! log JC* V logy* 0» 0.545306 9.736640 — 0.160483 n9.205429 1 0.545315 648 — 0.160607 5927 r, — 2 0.545310 644 0.160846 6410 s 0.545288 626 — 0.161019 6877 4 0.545245 592 — 0.161186 7327 5 0.545176 537 0.161345 7766 and for any given time T = TJ^ + r, we have x = — 0.081244 + a/r y = + 0.596075 + }fx } (492) Finally, to facilitate the computation of the hour angle fi — a=^ fx^ — (a (Art, 291), we prepare the values of /sij for each of the Greenwich hours. Thus, for T = 1*, we have From the Ephemeris, July 18, 1860, Sid. time at mean noon = 7* 46* 4* .03 Sid. equivalent of P mean t. = 1 9 .86 Greenwich sid. time = 8 46 13 .89 " <* " in arc, = 131* 33' 28".35 a =118 2 18 .16 Ml = 13 31 10 .19 Thus we form the following table, to which is also added for future use the value of the logarithm of fi' = the hourly difference of ii^ in parts of the radius; log Ai' = log 54002".15 sin 1" = 9.417986 /H Hourly diff. 0» 358" 31' 8".0 1 13 31 10 .2 2 28 31 12 .3 54002".15 8 43 81 14 .4 4 58 81 16 .6 5 78 81 18 .7 I proceed to consider the principal problems relating to the general prediction of eclipses, in which the preceding results will be applied. 456 SOLAR ECLIPSES. Outline of the Shadow on the Surface of the Earth. 298. To find the outline of the moon's shadow upon the earth at a given time, — This outline is the intersection of the cone of shadow with the earth's surface ; or, it is the curve on the surface of the earth from every point of which a contact of the son's and moon's limbs may be observed at the given time. Let T = the given time reckoned at the first meridian, and let a, rf, x, y, ?, and log i be taken firom the general tables of the eclipse for this time. Then the co-ordinates f , 37, f of any place at which a contact may be observed at the given time must satisfy the conditions (491), } (493) (I — tC) sin e = a: — ? {l — tC) cos § = y — 17 Let ^ = the hour angle of the point Z^ w=z the west longitude of the place; then we have ^ z=z fi — a =: /l^ — m and the equations (483) become S = p cos ^'sin 6 \ 7j =z p sin /cos d — /» cos f' sin d cob ^ > (494) C = /t> sin f ' smd-^ p cos / cos d cos * J The five equations in (493) and (494) involve the six variables f, 7;, ^, f ', «?, and §, any one of which may be assumed arbi- trarily (excluding, of course, assumed values that give impossible or imaginary results) ; then for each assumed value of the arbi- trary quantity we shall have five equations, which fully deter- mine five unknown quantities, and thereby one point of the re- quired curve. I shall take Q as the arbitrarj^ variable. In the present form of the equations (494), they involve the unknown quantity />, which being dependent upon ip' oannot be determined until the latter is found. This seems to involve the necessity of at first neglecting the compression of the earth, by putting p = Ij and after an approximate value of ^' has been found, and thereby also the value of />, repeating the computation. But, by a simple transformation given by Bessel, this double computation is rendered unnecessary, and the compression of OUTLINE OP THE SHADOW. 457 the earth is taken into account from the beginning. If ip is the geographical latitude, we have (Art 82) , cos ip . , sin cp (1 — ee) ^ cos f>= —, V- ^sm/= ^ ^'(l — ee sin' f>) ^'(l — ee sin' <p) in which log ee = 7.824409 log i/(l — ec) = 9.9985458 If we take a new variable ^ j, such that cos ip cos «Pi = —-' —, ^* |/(1 — ee sin* ^) we shall have //-I 1 N sin^T/(l— ee) |/(1 — ee sm' f>) or cos f^^p COS ^' p/(l — ee) sin ^^^ p sin f>' tany= ^^^> 1/(1 -ee) Hence the equations (494) become f = cos f , sin * ly ^ sin f J cos (Z |/(1 — ee) — cos ^^ sin d cos ^ C = sin ^j sin (Z p/(l — ee) + cos ^^ cos d cos »> Put p^ sin Jj =: sin d p^ sin ^, = sin d |/(1 — ^) ) /9j cos dj = cos d |/(1 — ee) />, cos d^ = cosd J ^ ^ The quantities />i, dj, />„ d,* ^^^7 ^^ computed for the same times as the other quantities in the tables of the eclipse, and hence obtained by interpolation for the given time. The factors /Oj and />, will be sensibly constant for the whole eclipse. We now have f = cos f J sin * ly = /9j sin f>j cos d^ — /9, cos ^ i sin d^ cos t> C = /o, sin ^, sin (f, + />, cos ^^ cos d^ cos i> Let us put Pi and assume (^„ so that ? + '?,• + C,« = 1 (496) 458 SOLAR ECLIPSES. or, which is equivalent, let us take the system f = cos f , sin * i^j = sin f J cos dj — cos f>j sin rf, cos ^ ^ (497) C, = sin f , sin d^ + cos ^ ^ cos d^ cos * The quantity f i differs so little from f that we may in practice substitute one for the other in the small term i^ ; but if theo- retical accuracy is desired we can readily find f when fj is known ; for the second and third of (497) give COB ^j COS * = — i^j sin d^ + C, cos d^ sin f^ = 1}^ cos dj + Ci sin d^ which substituted in the value of (^ give C = />, C, cos (d, - d,) - p, 71^ sin (rf, - d,) (498) Our problem now takes the following form. We have first the three equations (i-i:,)cose = y-^,,, V (499) f* + 'yi' + ^1* = 1 ) which for each assumed value of Q determine f , tj^j and f j. Then we have cos f>j sin * = f 'J COB ^j cos d = — i^j sin rfj + Ci cos d^ V (600) sin fj= i^j cos rfj + Ci sin d^ ) which determine <p^ and A Then the latitude and longitude of a point of the required outline are found by the equations tan sp = ^^"^' 0, = « — * (501) |/(1 -— ee) To solve (499), let ^ and y be found by the equations sin /9 sin T' = X — / sin Q = a then we have sm fi cos r = — = ^ $ =: sin iS sin ^^ + iCi sin Q ly^ = sin fi cos 7' + iC^ cos Q OUTLINB OF THE SHADOW. 459 where we have omitted /o^ as a divisor of the small term i!^^ cos Q, since we have very nearly p^ = 1. Substituting these values in the last equation of (499), we find Cj* = C08»/9 — 2i> Bin /9 cos (C — r) — (»Ci)' Neglecting the terms involving i* as practically insensible, this gives Ci= ± [cos fi — I sin fi cos (Q — r)] In order to remove the ambiguity of the double sign, let us put Z = the zenith distance of tJie point Z(Art. 289) ; then, since & = fi — a is the hour angle of this point, we have cos Z == sin f sin d -f cos f cos d cos ^ which by means of the preceding equations is reduced to C08 2 = c. /., ^ (503) sm ^j Hence cos Z and (^^ have the same sign. But, in order that the eclipse may be visibk from a point on the earth's surface, we must, in general, have Z less than 90° ; that is, cos Z must be positive, and therefore f ^ must be taken only with the positive sign. The negative sign would give a second point on the surface of the earth from which, if the earth were not opaque, the same phase of the eclipse would also be observed at the given time. In fact, every element of the cone of shadow which intersects the earth's surface at all, intersects it in two points, and our solution gives both points. K we put i cos (Q^r) sin 1" we have Cj = cos fi — sin /9 sin e (604) or, with sufficient accuracy, C, = cos (fi + e) (505) Thus, fi and y being determined by (502), f j is determined by (504) and (505) : hence also f and ij^ by the equations 460 80LAR ECLIPSES. The problem is, therefore, fully resolved ; but, for the conve- nience of logarithmic computation, let c and C be determined by the equations '^"°^='' 1(507) c cos (7 = Ci J then the equations (500) become cos sp, sin 1^ = ^ 1 cos f J cos d = (? cos (C + (Zj) > (508) sin ^j= c sin ((7 + d^) ) The curve thus determined will be the intersection of the penumbral cone, or that of the umbral cone, with the earth's surface, according as we employ the value of I for the one or the other. 299. The above solution is direct, though theoretically but approximate, since we have neglected terms of the order of t*. It can, however, readily be made quite exact as follows. We have, by substituting the values of ^^ and tJj^ in (498), and neg- lecting the term involving the product isin({^ — d^), which is of the same order as i*, C = /o, cos (/? + «) — /», sin /9 cos j^ sin (d^ — d^) and, putting e' = (tfj — d^) cos 7' we have, within terms of the order i', C = />, cos (/5 + c + e') (509) The substitution of this value of ^ in the term if involves only an error of the order t^, which is altogether insensible. The exact solution of the problem is, therefore, as follows. Find ^ and Y for each assumed value of Q, by the equations sin fi sin ]r =^ oc — I sinQ = a . ^ y ^ cosO , sm 13 cos r = — —= Pi Pi then e and e' by the equations sm 1" OUTLINE OF THE SHADOW. 461 Find ^' and y' by the equations sin /?' sin / = a + i>, cos (/9 -f- e + c') sin Q = $ . ^- , 1 . t>, cos (^ + c + c') cos Q sin /9' cos •/= b + -^^-? ^^—5- — -!- — ^ — = 19, Pi then we have, rigorously, Cj = cos /9' and these values of f , i^j, and (^^ may then be substituted in (500), which can be adapted for logarithmic computation as before.* 300. It remains to be determined whether the eclipse is begin- ning or ending at the places thus found. A point on the earth's surface which at a given time T is upon the surface of the cone of shadow will at the next consecutive instant T -{- dT be within or xcithout the cone according as the eclipse is beginning or ending at the time T; the former or the latter, according as the distance A = i/[(a: — f)* + {y — yjf] becomes at the time T + dT less or greater than the radius of the shadow I — i^. In the case of total eclipse I — i^ is a negative quantity, but by comparing J* with (I — i(^y we shall obtain the required criterion for all cases ; and, therefore, the criterion of beginning or ending, either of partial or of total eclipse, will be the negative or positive value of the difterential coeflSicient, relatively to the time, of the quantity or the negative or positive value of the quantity '^ ^\dT dTr^ ^'\dT dT} ^ ^\dT dTl * In this problem, as weU as in most of the subsequent ones, I haye not followed Besskl's methods of solution, which, being mathematically rigorous, though as simple as such methods can possibly be, arc too laborious for the practical purposes of mere prediction. As a refined and exhaustive disquisition upon the whole theory, Bessel's Analyse der Finstemuie, in his Aitronovmche Untertuchungen, stands alone. On the other hand, the approximate solutions heretofore in common use are mostly quite imperfect ; the compression of the earth, as well as the augmentation of the moon's semidiameter, being neglected, or only taken into account by repeating the vhole computation, which renders them as laborious as a rigorous and direct method. I haye endeaTored to remedy this, by so arranging the successive approximations, when these are necessary, that only a small part of the whole computation is to be repeated, and by taking the compression of the earth into account, in all cases, from the commencement of the computation. In this manner, even the first approxima- tions by my method are rendered more accurate than the common methods. 462 SOLAR ECLIPSES. where we omit the insensible variation of i. For brevity, let as dx dv write x\ y\ &c. for -77=7, -r—-, &c. and denote the above quantity by P; then, after substituting the values of x — f = (Z — i f ) sin §, jf — ij = {I — i^) cos Qj we have P= L [(x'- e')8^n « + (y^- V)co8C - (/'- tC')] in which L=l — i(^. K we put P' = (a/ — e') sin e + (y — ,') cos c — (r - tr) (6io) we shall have The quantity P will be positive or negative according as L and P' have like signs or different signs. For exterior contacts, and for interior contacts in annular eclipse, L is positive (Art. 293), and hence for these cases the eclipse is beginning or ending according as P' is negative or positive; but for total eclipse, L being negative, we have beginning or ending according as P' is positive or negative. We must now develop the quantity P'. Taking one hour as the unit of time, x', y', 1% f ', 7j\ ^', will denote the hourly changes of the several quantities. The first three of these may be derived from the general tables of the eclipse for the given time; but f ', r/, ^' are obtained by differentiating the equations (494), in which the latitude and longitude of the point on the earth's surface are to be taken as constant. Since 1? = ft — a>, we shall have —p=f = -j^ ; and hence, putting we find y=,*^8inr d'= — sinr dT dT r' = IX p cos f>'co8 ^ = pJ ( — yj Bin d + C cos d) = fi' [ — y sin d + C cos d -]- (I — iC) sin d cos Q] y = ju' $ sin d — d'Z = /A Ix sin d — (I — I'C) sin d sin Q] — d'C C'=— /t'eco8rf + d'li = p! [—X coBd + (I — tC) cos 4 sin Q] + d' [t/ -- (I — tC) cos Q} OUTLINE OF THE SHADOW. 463 Substituting these values in (510), and neglecting terms involving I* and id' as insensible, we have P'=z a' — y cos Q + </ smQ — C (m' cos ^ sin Q — d' cos Q) in which a', 6', and c', denote the following quantities: a'= — r — ju' ix cos d "j h'= — y'+ii'x sin d V (511) (/ = xf -{- //y sin d -\~ fi' il cos d ) The values of these quantities may be computed for the same times as the other quantities in the eclipse tables, and their values for any given time will then be readily found by interpo- lation. For any assumed value of Qj therefore, and with the value of Z found by (509), the value of P' may be computed, and its sign will determine whether the eclipse is beginning or ending. In most cases, a mere inspection of the tabulated values of a', 6', and c', combined with a consideration of the value of Qy will suffice to determine the sign of P' ; but when the place is near the northern or southern limits of the shadow, an accu- rate computation of P' will be necessary; and, since other appli- cations of this quantity will be made hereafter, it will be proper to give it a more convenient form for logarithmic computation. Put csinJS7 = y fBmF=d' lr512^ eQO%E=cf /cos^=ju'cos d j^ ^ then we have P' = a' + « sin (C — -^ — C/ sin (C — F) (513) Since a' and -Pare both very small quantities, and a veiy precise computation of P' will seldom be necessary when its algebraic sign is alone required, it will be sufficient in most cases to neglect these quantities, and also to put f j for f , and then we shall have the following simple criterion for the case of partial or annular eclipse : If e sin (Q — ^) < Ci/ sin §, the eclipse is beginning. If e sin (Q — i?) > Cj/ sin Q, the eclipse is ending. For total eclipse, reverse these conditions. 801. In order to facilitate the application of the preceding as well as the subsequent problems, it is expedient to prepare the values of d^^ log p^^ d^, log />„ a', fc', c', e, -E, /, 2^, and to arrange them in tables. '*7 » *■ "XJ 3b 3«8t wilii^ jr* 4- ?•• i. 3 ^ L. S^ -' «i 4;> 9(> i** ■*.:•»< r .JUi. lu** <»•.'. -• i-I^r^tV" l«!*Sf- i^J. • TT- - 1 --11 ttTTTTtt. Aif* fram _ I ~ '.^GZiH idoi* ic 4iC» Jl.'"n-.'* i' •• '' " »• , ^■•l' —1,* ""li"**^ ""J" l' ■ iv*^ •• - ■ i^r^v n_ ^r n-iTTif riTCSB-fc — - ^ r t « 1 it»«<-ii _ 1 •»•♦■♦_ T:-<r vaij"* of '. K. f. F. i'ffT rjiirrtr r:#*»w.^*v. dt:^iuc>rd from TL<r--r v^; j*'*- *>f 6' ajid c'. a;.d from (f'= — 25 ".o sin l'\bv lolik, OUTLINE OF THE SHADOW. 465 E log* F log/ 0» 4° 33' 21" 9.8019S9 — 0' 1' 44" 9.388244 1 9 22 25 .795965 « 264 2 14 17 17 .793034 « 285 8 19 13 48 .793255 « 805 4 24 7 46 .796604 u 826 5 28 55 7 .802923 « 347 302. To illustrate the preceding formulse, let lis find some points of the outline of the penumbra on the earth's surface at the time T= 2* 8* 12*. For this time, we have a; = — 0.00672 ^ogp^= 9.99873 log i = 7.66287 y = + 0.57409 d^= 21<» 0' 45" I = + 0.53673 ti^= 30 34 13 Let us find the points for Q = 50° and Q = 800°. The com- putation may be arranged as follows : Q By (502) : Hence by (504) : By (506) : By (507) : log log a = sin /9 sin y 6 = sin ^ cos Y r P By (505) : log C, = log cos (/9 + e) iCiSin Q iZi cos Q 17, = log c sin C Cj = log c cos C log^ C+rf, log cos f J sin i> log cos ^j cos »^ log tan d log cos f J log sin ip^ log tan ip^ log i/(l — ee) log tan f 9 Vol. L*30 By (508) : log ? logc cos (C+ rfj) logc sin (C+ <^i) = 60« 8OO0 0.41788 h 0.45810 + 0.22975 - 0.30662 61<» ir 52" 56<» 12' 16" 28 28 52 33 27 7 5 43 — 6 59 28 23 9 33 20 8 9.94437 9.92193 h 0.00310 — 0.00333 - 0.00260 + 0.00192 0.41478 + 0.45477 + 0.23235 + 0.30854 9.36614 9.48931 9.94437 9.92193 9.95901 9.94969 14^ 47' 39" 20^16' 9" 35 48 24 41 16 54 n9.61782 9.65779 9.86803 9.82560 n9.74979 9.83219 9.92764 9.90803 9.72620 9.76908 9.79856 9.86105 9.99855 9.99855 9.80001 9.86250 — 29<» 20' 20" 34^*11' 46" 59 54 33 356 22 27 32 15 3 36 4 40 466 SOLAR ECLIPSES. To find whether the eclipse is beginning or ending at these places, we have, from the table on p. 465, for T=^ 2* 8"* 12*, lege 9.7931 E 14^ 58' Q E 85 2 285<> 2' log e sin (C — E) 9.5521 n9.7780 log/ 9.3883 log Ci/ sin Q 9.2170 n9.2477 At the first point, therefore, we have e sin (Q — -E) > fj/sin §, and the eclipse is ending. At the second point, we have 6 sin(§ — E) < Qy^f&in Q^ and the eclipse is beginning. Rising and Setting Limits. 303. To find the rising and setting limits of the eclipse. — ^By these limits we mean the curves upon which are situated all those points of the earth's surface where the eclipse begins or ends with the sun in the horizon. It will be quite sufficient for all practical purposes to determine these limits by the condition that the point Z is in the horizon. This gives in (503) cos^=0, or Ci = 0, and, consequently, by (496), we have as the condition which the co-ordinates of the required points must satisfy. Now, let it be required to find the place where this equation is satisfied at a given time T. Let x and y be taken for this time, then we have, by putting Ci= in (499), ? sin Q = X — ^ ? cos § = y — ly Let m sin Jtf" = a: m cos M =^y then, from the equations ? sin Q = m %\n M — p sin y I cos Qz=vi cos M — p cos y we deduce, by adding their squares, Z* = m' — 2mp cos (3f — f) ^ p* 2 sin«}(Jtf— y) = 1 — cos (3f — r) = ^— (^ — 1^)' 2mp I? sin z' = f p cos r ^=^ ] (516) } (516) RISING AND SETTING LIMITS. 467 If then we put k=^ M — r? we have 8in i>l = dz ^p + m--;2j^-m+j>)1 (517) Y -= M :^k Hence we have m which \X may always be taken less than 90°, but the double sign must be used to obtain the two points on the surface of the earth which satisfy the conditions at the given time. In this formula, m, 3/, and I are accurately known for the given time, but p is unknown. It is evident, however, from (514) and (515), that we have nearly J» = 1, and this value may be used in (517) for a first approximation. To obtain a more correct value of 7*, let us put f = sin y'\ then, by (514), we have 1^1= cos 7^, and, consequently, since tj^ p^jj^, p Bin y = sin / p cos Y z=z p^ cos f tan •/ z= p^ tan y sin/ p.cos/ } (518) p = — — — = *-^ 2- I sin Y cos Y and with this value of p the second computation of (517) will give a very exact value of 7*. With this second value of 7* a still more correct value of p could be found; but the second approxi- mation is always sufficient. With the second value of 7*, therefore, we find the final value of f by the formula tan / = /Oj tan y and then, substituting the values f = sin y\ t^^ = cos y^, ^j == 0, in (500), we have, for finding the latitude and longitude of the required points, the formulse cos f J sin ^ = sin / COB f , cos & = — cos / sin d^ sin ^j = cos / cos rf, (519) a> = /I, — 1* tan f == ^ — 1/(1 — ee) In the second approximation, we must compute X and y by (517) separately for each place. 468 SOLAK ECLIPSES. 304. Tlie sun is rising or setting at the given time at the places thus deteniiined, according as ^ (which is the hour angle of the point Z) is between 180° and 360° or between 0° and 180°. To determine whether the eclipse is beginning or ending, we may have recourse to the sign of P' (513); and it will usually be sufficient for the present problem to put both a' and f = in that expression, and then the eclipse is beginning or ending according as sin {Q — E) is negative or positive. Now, by (516), we find I sin (Q — ^) = m sin (ilf — E) — p sin (r — JS) Hence, for points in the rising or setting limits, If m sin (ibf — E) <Cp sin (y — E), the ecHpse is beginning, If m sin (M — E) ^ p sin (jr — E), the ecHpso is ending. 305. In order to apply the preceding method of determining the rising and setting limits, it is necessary first to find the extreme times between which the time T is to be assumed, or those limits of T between which the solution is possible. The two solutions given by (517) must reduce to a single one when the surface of the cone of shadow has but a single point in common with the earth's surface, — i.e. in the case of tangeney of the cone and the terrestrial spheroid. Now, the two solutions reduce to one only when ^ = 0, and both values of y become == 3f; but if k = 0, the numerator of the value of sin iX must also be zero; and hence the points of contact are determined by the conditions I -\- VI — p =z and I — m -]- p = or bv the conditions VI = p -\- I and m = p — I There may be four cases of contact, two of exterior and two of interior contact. The two exterior contacts are the first and last, or the beyitudvg and the end of the eclipse generally ; the axis of the shadow is then without the earth, and therefore we must have for these cases m = \^dr + y' = p -{- L The first interior contact corresponds to the last point on the earth's surface where the eclipse ends at sunrise ; the second, to the first point where it begins at sunset. But these interior RISING AND SETTING LIMITS, 469 contacts can occur only when the whole of the shadow on the principal plane falls within the earth, and for these cases, there- fore, we must have m = p — L For the beginning and end generally we have, therefore, by (515), (p -{- 1) »\n M =z X (p -f- I) cos M=y Let The the time when these conditions are satisfied, and put in which Tq is the epoch of the eclipse tables, for which the values of x and y are x^ and y^. Then, x' and ^' being the mean hourly changes of x and y for the time T, we have } (520) Puttmg m^ sin M^ = x^ n sin N = x' m^ cos M^ = yo ^ cos N=\f the above conditions become (l> -f I) sin M= Mq sin M^ -\-t ,n»\n N (p -|- cos M= hIq cos M^-^r ,n cos N whence (p + I) sin (3f — iV) = m^ sin (M^ — iVT) iP + cos (Jf — iVT) = m^ cos (M^ — N) +nT so that, if we put JK* — iV = -v^, we have sin 4, = ^o sin (M, - N) p + l r=P±Icos^-'^cos(M,-N) ) (^21) T= T^ + T in whicn cos -J/ may be taken ^vith cither the negative or the positive sign; and it is evident that the first will give the beginning and the second the end of the eclipse generally. For the two interior contacts we have Bin 4 = >!!>j}.( -K - ^) f~' } (522) T = COS 4* ^ cos (M^ — iV ) 470 SOLAR ECLIPSES. These interior contacts cannot occur when p — Ms less than 7«o sin {Mq — N), which would give impossible values of sin ^. In these formulae we at first assume 2> = 1, and, after finding ail approximate value of ^'j we have, by (517), in which x = 0, 2' =^ M, and in the present problem M= N+ '^i therefore ^ = JVr+4, (523) with which p is found by (518), and the second computation of (521) or (522) will then give the required times. We niusst employ in (523) the tvvo values of ^^ found by taking cos ij^ with the positive and the negative sign ; and therefore different values of p wU be found for beginning and ending, so that in the second approximation separate computations will be necessary tor the two cases. In the first approximation the mean values of x', y', and / may be used, or those for the middle of the eclipse. With the approximate values of r thus found, the true values of r', /, and I for the time T= Tq + r may be taken for the secoud approximation. After finding the corrected value of ^'j we then liave also the true value of y = N + '\l/ for each point, and hence also the true value of y^ by (518), ^vith which the latitude and longitude of the points will be computed by (519). For the local appan*iit time of the phenomenon at each place we may take the value of ?? in time, which is very nearly the suif s hour angle. 306. When the interior contacts exist, the rising and Bcttiiiir limits form two distinct enclosed curves on the earth's surtaoe. If wc denote the times of beginning and ending generally, de- termined by (521), by T^ and T^, and the times of interior oon- tiict, determined by (522), by y/ and 71', a series of points on the risins: limit will be found bv Ait. 303, for a series of time? assumed between 7\ and 7^/, and points of the setting limit tor times assumed between T^^ and T^, When the interior contacts do not exist, the rising and settiii*: limits meet and form a single curve extending through the whole eclipse. The form of this curve may be compared to that of the figure 8 much distorted. A series of points upon it will be found by assumiug times between 7\ and T^. 307. Example. — Let us find the rising and setting limit:* of the e-lipse of July 18, 1860, RISING AND SETTING LIMITS. 471 Ih'st — To find the beginning and ending on the earth gene • rally, we have for the assumed epoch Tq = 2\ page 455, m^ Bin M^ = x^ = — 0.081244 m^ COB M^ = yo = + 0.596075 which give log m^ = 9.77930 M^ = 352° 14' 19" log m^ sin (Jf^ — N) = n9.73938 n sin -Ar= a/= + 0.5453 n cos iV= 2^= — 0.1608 n log n = 9.75474 iV=106<»25'.8 C08(ifp—iV) = — 0.4336 For a first approximation, taking 2? = 1, we find, by (521), p + l = 1.5367 log sin ^ p + l n cos ^ = T n9 .5528 m 2».525 m. j;__^coB(Jtfp— iVr) = + 2.434 n Approx. beginning Tj end r. u 23».909 4.959 (Jnly 17) (July 18) Taking cos ij/ negative for beginning and positive for ending, we have then, by (518) and (523), log tan y log Pi log tan f log sin / log sin ^^ logi> P I p + l For the above computed times we further find Beginning. End. 200<» 55'.4 339<» 4'.6 307 21.2 85 30.4 n0.11732 1.10466 9.99873 9.99873 nO. 11605 1.10339 9.89985 9.99865,5 9.90032 9.99866,3 9.99953 9.99999 0.99892 0.99998 0.53687 0.53640 1.53579 1.53638 log jf logy' log n sin N log n cos JV log n 9.73664 9.73654 719.20538 n9.20774 9.75467 9.75477 106° 23' 50" 106° 29' 8" 472 SOLAR ECLIPBES. For a second approximation, therefore, recomputing (521), we now find and by (618) : log sin 4/ log cos 4 T log tan / n9.65316 n9.97032 23*.9098 200<' 56' 27" 307 20 17 n0.11629 n9.55269 9.97039 4*.9587 339° 4' 68^ 85 34 6 1.10942 Then, for the latitude and longitude of the points, we have. by (519), th — *== w 9 2r r42" 357 9 57 254 38 57 102 31 34 38 34 20*^ 59' 33" 72 54 8 91 35 43 341 18 25 4 9 46 Therefore the eclipse begins on the earth generally on July 17, 23* 54'".5 Greenwich mean time, in west longitude 102® 31' 0" and latitude 34° 38' 34", and ends July 18, 4^ 57"*.5 in longitude 841° 18' 25" and latitude 4° 9' 46". It is evident that for practical purposes the first approximation, which gives the times within a few seconds, is quite sufficient, especially since the effect of refraction has not yet been taken into account. (See Art. 327.) Secondly, — We now pass to the computation of the cun^e which contains all the points where the eclipse begins or ends at sun- rise or sunset. In the present example, this curve extends* through the whole eclipse, since we have m^ sin {31^ — N) > 1 — /: hence the required points will be found for Greenwich times assumed between July 17, 23*.91 and July 18, 4*.96. Let us take the series r, 0*, 0*.2, 0*.4, 0*.6, 0*.8 4».6, 4».8 The computation being carried on for all the points at once, the regular progression of the corresponding numbers for the suc- cessive times furnislies at each step a verification of its correct- ness. To illustrate the use of the formula*, I give the computa- tion for T~ 2*.0 nearly in full. For this time, we find, from p. 454 and p. 464, X = in sin M= — 0.08124 y =^ vi cos M = + 0.59608 I = 0.53675 rf,= 21o0'49" log /Oj = 9.99873 RISING AND SETTING LIMITS. 473 and hence M = 352<' 14' 21" log m = 9.77931 m == 0.60160 Then, by (517), taking ;? = 1, we have ar. CO. log 4tmp 9.61863 Ij^m—p = 0.13835 log 9.14098 l^rn+p = 0.93515 log 9.97088 ;i = 26° 49' log 8in» } k 8.73049 With this first approximate value of X we find the value of p for each of the two points, by (518), as follows : log tan Y log /o, tan Y = log tan / O cos r loff " — = loff p ^ cos /' ^^ p 19° 3' 9.53820 9.53693 9.99887 0.99740 325° 25 9.83849 9.83722 9.99914 0.99802 Repeating (517) with these values of p : ar. CO. log 4 mp log (I + m^ p) log (Z — m + 2?) log sin' } X ±iX M ± X = Y log tan Y log tan / Hence, by (519), For T= 2*. (p. 455), fi, fi^—^z= at Local app. time = ^ in time, 9.61976 9.14907 9.96967 8.73850 + 27° 4' 4" 19 18 25 9.54448 9.54321 9.61949 9.14715 9.96996 8.73660 — 27° 0' 26" 325 13 55 n9.84148 n9.84021 135° 45' 4" 28 31 12 252 46 8 61 52 35 9* 3-.0 Sunset. 242° 36' 45" 28 31 12 145 54 27 50 13 46 16» 10-.45 Sunrise. To find whether the eclipse is beginning or ending at these points, we have, from p. 465, and by Art. 304, E 14° 17' log m sin (M — E) n9.3538 n9.3538 \ogp sin (r — E) 8.9406 n9.8772 Beginning. Ending. In the same manner are found the results given in the following table : 474 SOLAR ECLIPSES. SOLAR ECLIPSE, July 18, I860.— RISING AND SETTING LIMITS. Greenwich Latitude. Long. W. from Greenwich. Local App. Time. Mean Time. ^ u ^ o*.o + 44*^ 27' 110° 35' 16* 31-7 Begins at Sunrise. .2 52 34 121 33 15 59 .8 n u .4 58 1 132 21 15 28 .7 l( it .6 62 10 144 2 14 53 .9 u u .8 65 21 157 6 14 13 .7 a a 1.0 67 36 171 46 13 27 .0 a " 1 .2 68 49 187 56 12 34 .4 u a 1 .4 68 58 204 56 11 38 .3 u Sunset. ' .6 67 55 221 51 10 42 .7 a u .8 65 37 237 54 9 50 .5 u u 2.0 61 53 252 46 9 3 .0 <( li .2 56 16 266 33 8 19 .9 it a .4 48 5 279 17 7 41 .0 a 11 .6 37 15 290 36 7 7 .7 u It .8 25 6 300 12 6 41 .3 ti a ■ 3 .0 13 36 308 12 6 21 .3 n 4< .2 + 3 59 315 6 6 .1 (C it .4 — 3 24 320 50 6 54 .8 11 tt .6 8 43 325 53 5 46 .5 tt ft 1 .8 12 14 330 17 5 41 .0. u tt 4.0 14 11 334 4 5 37 .8 (4 ii .2 14 48 337 19 5 36 .8 <i i( .4 14 6 340 2 5 38 .0 Ends il .6 11 56 342 9 5 41 .5 (( n ; .8 — i 82 343 25 5 48 .4 a (( 0.0 + 25 45 99 10 17 17 .4 Begins at Sunrise., .2 20 1 99 33 17 27 .9 a U I .4 17 16 101 22 17 32 .6 u « .6 10 7 103 52 17 34 .6 Ends it .8 16 17 17 46 106 56 110 34 17 34 .3 17 31 .8 t» iC 1 .0 u i» .2 20 42 114 50 17 26 .7 ii kk .4 25 17 119 57 17 18 .3 it ii .6 31 45 126 14 17 5 .2 it ti .8 40 134 15 145 54 16 45 .0 u •' '; 2.0 50 14 16 10 .5 If «* .2 60 21 168 47 15 10 .9 u k* .4 07 27 191 43 13 31 .2 u i< .6 ()S 55 224 18 11 32 .9 li Sunset. .8 66 27 249 7 10 5 .6 u t( CURVE OF MAXIMUM IN THE UORIZON. 475 SOLAR ECLIPSE, July 18, 1860.— RISING AND SETTING LIMITS.— (Con/mu^rf.) Greenwich Mean Time. Latitude. Long. W. from Greenwich. (J Local App. Time. 3*.0 .2 .4 .6 .8 + 62*^ 43' 58 44 54 42 50 35 46 21 265*' 37' 277 27 286 49 294 47 301 53 9* 11-.6 8 36 .3 8 10 .8 7 51 .0 7 34 .6 Ends at Sunset. <( ii ({ ti 4.0 2 .4 .6 .8 41 55 37 10 31 57 25 55 18 11 308 26 314 40 320 43 326 48 333 18 7 20 .3 7 7 .4 6 55 .2 6 42 .9 6 28 .9 ii ii ii (. it Ii il ii ii a These points being projected upon a chart (see p. 504), the whole curve may be accurately traced through them. It will be seen that the method of assuming a series of equidistant times gives more points in those portions of the curve where the curvature is greatest than in other portions, thus facilitating the accurate delineation of the curve. This advantage appears to have been overlooked by those who have preferred methods (such, for example, as IIansen's) in which a series of equidistant latitudes is assumed. 308. The preceding computations have been made for the penumbra ; but we may employ the same method to determine the rising and setting limits of total or annular eclipse by employing in the formulae the value of I for interior contacts. These limits, however, embrace so small a portion of the earth's surface that they are practically of little interest Curve of Maximum in the Horizon. 309. To find the curve on which the maximum of the eclipse is seen at sunrise or sunset. — When a point of the earth's surface whose co-ordinates are f, tj, and ^ is not on the surface of the cone of shadow, but at a distance J from the axis of the cone, we have the conditions (485), J sin ^ = X — ^ J cos § = y — 1? } (524) 476 SOLAR ECLIPSE^. The amount of obscuration depends upon the distance by which the place is immersed within the shadow, that is, upon the distance i— J, i being the radius of the shadow on the parallel phme at the distance ^ from the principal plane. For the maximum of the eclipse, therefore, we have the condition (IL ___^-f__Q dT dT Difterentiating the above equations relatively to the time, and denoting the derivatives of a:, ^, &c. by accents, as in Art. 300, we have ^"^ sin Q — Jco8Q.-^ = .r'— r dT " " dT ^^ cos (2 + J sin §. i§- = y'— r/ dT ^ ' ^ dT which give "^"^ = (x!— e') sin Q + (y'— V) cos Q dT The equation L^=l — i^ gives dT and, therefore, V— i:'— (a/— ^') sin Q — (/— r/) cos § = (525) or, by (510), P' = (526) This is, therefore, the general condition which characterize:? the maximum of the eclipse at a given time. In the j)re!*ent problem we have also the condition that the sun is in the horizon, for which we may, as in Art. 303, substitute the condition ^, — 0. Since, however, the instant of greatest obscuration is not subject to any nice observation, a very precise solution of the problem is quite unimportant, and we may be satisfied with tlie approxi- mate solution obtained by supposing 1^ = 0^ and at the same time neglecting the small quantity a' in P'. The condition (526) will then be satisfied when in (513) we have that is, when Q== E or Q = 180* + E CURVE OF MAXIMUM IN THE HORIZON. 1, for any given time, the conditions (524) become -S ± J sin £ ^ a; - ± J cos £ = y - k with the condition t determine the required points of our curve. The angle S bre known for tlie given time, being directly obtained from Vbulated valuea, but J is unknown. Putting, aa in the ndjug problem, meia M = m cos M = p COB}- = ± d ain E =^ m eia M - ± J cos E = m COB M - afore, putting 4- m sin (jtf — E) — p aia(y — E) w cos (M — £) — pcoa(jr — E} — ^, we have nain jM—E) ± J ^ m COB (Jf — E) — pco6^ B first of these equations will ^ve two values of i|/, since we IT take cos n^, with the positive or the negative sign ; but, as r thoso [tlaces satisfy the problem which are actually icitkiti ^ bIiuiIow, we must have J < i, or, at least, J not greater than I. jit value of il- which would give J> I must, therefore, be ■ludcd : BO that in general we shall have at a given time but R'Bointion. ^urill be iiuitc accurate enough, considering the degree of ion above assigned, to employ in (527) a mean value of p, ■e p fulls bet^vecn /), and unity, to take log p = ^ log /),. if we wiflh a more correct value, wo have only to take = * + E (528) I then find p as in (518) ; after which (527) must bo recom- A 478 SOLAR ECLIPSES. Having found the true value of ->// by (527), and of y by (528), we then have f by the equation tan /^ = /»! tan x and the latitude and longitude of each point of the curve by (519). The limiting times between which the solution is possible will be known from the computation of the rising and setting limits, in which we have already employed the quantity m sin (wS/— JF); and the present curve will be computed only for those times for which m sin (Jfef — E) < I, These limiting times are also the same as those for the northern and southern limiting curves, which will be determined in Art. 313. 310. The degree of obscuration is usually expressed by the fraction of the sun's apparent diameter which is covered by the moon's disc. "When the place is so far immersed in the penumbra as to be on the edge of the total shadow, the obscuration is total ; in this case the distance of the place from the edge of the penumbra is equal to the absolute difference of the radii of the penumbra and the umbra, that is, to the algebraic sum JL + Lp ii denoting the radius of the umbra (which is, by Art. 293, negative) ; but in any other case the distance of the place within the penumbra is L — d: hence, if D denotes the degree of obscuration expressed as a fraction of the sun's apparent diameter, we shall have, very nearly. This formula may also be used when the eclipse is annular, in which case L^ is essentially positive ; and even when J is zero, and the eclipse consequently central, the value of D given by the formula will be less than unitj', as it should be, since in that case there is no total obscuration. In the present problem we have B ■= ^-^^ (529») l + h in which I and ?i are the radii of the penumbra and umbra on the principal plane, as found by (488). Example. — In the eclipse of July 18, 1860, compute the curve on which the maximum of the eclipse is seen in the horizon. CURVE OF MAXIMUM IS THE HORIZON. 479 In the computation of the rising and setting limits, the quantity m sin {M— E) was less than unity only from 7*= 0*.6 to r= 4*.2 : so that the present curve may be computed for the series of times 0*.6, 0*.8 4*.0, 4*.2. For an approximate computation we may take \ogp = Jlog p^= 9.9994, and employ only four decimal places in the logarithms throughout The computation for 7* = 2* is as follows. For this time we have already found (p. 473) Hence, by (527), E M^E log m sin (M — E) log2> log sin ^ log cos ^ logjp cos 4* log m cos (3f — E) m cos {M — E) jpcos ^ J 9.7793 362<' 14'.4 14 17.3 337 57.1 n9 3538 9.9994 n9.3544 9.9886 9.9880 9.7463 + 0.5575 + 0.9727 0.4152 Here, if cos o^ were taken with the negative sign we should find J= 1.5302, which is greater than I. Taking it, therefore, with the positive sign only, we have log p^ = 9.9987 * + E=:r log tan z' log tan / — 13« 4'.3 + 1 13. 8.3271 8.3258 with which we find, by (519), w 9 App. time =: tl^ in time 176*' 87'.2 28 31.2 211 54 69 1 11» 46-5 Sunset. To express the degree of obscuration according to (529*) we have, taking the mean values of I and Ix (p. 454), I = 0.5866 I, = — 0.0092 i — J = 0.1214 bJ^^^^ 0.28 I + I,= 0.5274 0.5274 In the same manner all the following results are obtained : 480 80LAR ECLIPSES. SOLAR ECLIPSE, July 18, I860.— CURVE OF MAXIMUM OP THE ECLIPSE IN THE HORIZON. Greenwich Mean T. Latitude. Long. W. Arom Greenwich. u App. Local Time. Degree of Obsourktioa. D 0*.6 0.8 1.0 1.2 1.4 + 24° 44' 37 47 47 3 54 31 60 38 107° 41' 117 47 127 49 139 1 152 24 17* 19-3 16 50 .9 16 22 .8 15 50 .0 15 8 .5 0.30 .76 .97 .74 .56 1 .6 1 .8 2.0 2.2 2.4 65 20 68 16 69 1 67 34 64 20 169 189 16 211 54 233 32 251 42 14 14 .1 18 5 .0 11 46 .5 10 31 .9 9 31 .3 .41 .31 .23 .18 .17 2 .6 2.8 3.0 3.2 3.4 59 55 54 41 48 52 42 35 35 49 266 11 277 50 287 31 295 66 303 30 310 S3 317 22 324 15 331 14 8 45 .3 8 10 .8 7 44 .0 7 22 .4 7 4 .1 .17 .21 .28 .37 .50 3.6 3 .8 4.0 4.2 28 28 20 21 + 11 2 45 6 47 .9 6 32 .6 6 17 .2 6 1 .1 .67 .89 .87 .48 Northern and Southern Limiting Curves. 811. To find the northern and southern limits of the eclipse on the earth's surface. — These limits arc the curves in wliicli are situated all the points of the surface of the earth from wliich only a sin»rle contact of the discs of the sun and moon can be observed, the moon appearing to pass either wholly south or wholly north of the sun. They may also be defined as curves to which the out- line of the shadow is at all times in contact during it8 progre;?3 across the earth. The solution of this j)roblem is derived from the consideration that the sim])le contact is here the maximum of the eclipse, so that we must have, as in (526), P' = and consequently, by (513), a' + e sin ((? — JE:) = C/sin (C — F) (530) NORTHERN AND SOUTHERN LIMITS. 481 For any given time T^ therefore, we are to find that point of the outline of the shadow on the surface of the earth for which the value of Q and its corresponding f satisfy this equation. This can be effected only indirectly, or by successive approxima- tions. For this purpose, we must know at the outset an approxi- mate value of Q; and therefore, before proceeding any further, we must show how such an approximate value may be found. We can readily determine sufficiently narrow limits between which Q may be assumed. For this purpose, neglecting a' in (530), as well as F^ which are always very small, we have, approximately, 6sin(Q — JB)==C/8in Q The extreme values of f are C = ^ and C = !• The first gives sin (§ — -&) = 0, and therefore for a first limit we have Q=zE or Q = 180° -f E The second gives 6 8in(Q — ^)=/8in Q whence Put tan ( Q — } JB) = ^-i^ tan } S e 4- f tan ^ = — ^-^tan iE then the equation tan {Q — J-^ = tan o^ gives for our second limits Q=iE+'^ or C = 180° + J^+4' To compute o^ readily, put tan y = — then ' ) (^«1) tan 4 = tan (45** + v) tan }-B and Q is to be assumed between E and i E -{- -^ or between 180° + JEand 180° + i E + -^ Vol. L— 31 482 SOLAR ECLIPSES. These limits may be computed in advance for the principal hours of the eclipse from the previously tabulated values of -E!, e, and /, and an approximate value of Q may then be easily inferred for a given time with sufficient precision for a first approximation. When the shadow passes wholly within the earth, there are two limiting curves, northern and southern. For one of these Q is to be taken between E and J -E + 4^ 5 for the other, between 180° + E and 180° + J JS; + o^. Since £J is always an acute angle, positive or negative, it follows that when Q is taken between U and I E+ '\^jit& cosine is in general positive, while it is nega- tive in the other case. The equation jy = y — {I — ?C) ^^ Q shows that rj will be less in the first case and greater in the second, and hence the values of Q between E and \ E -\- '^ belong to the southern limits and the values of Q between 180^ + E and 180° -\- \ E -\- '^ belong to the northern limit. There is only one limit, northern or southern, when one of the series of values of Q would give impossible values of Z i*^ ^^ computation of the outline of the shadow by Art. 298. But when the rising and setting limits have been determined, the question of the 'existence of one or both of the northern and southern limits is already settled ; for if the rising and setting limits extend through the whole eclipse in north latitude, only the southern limiting curve of our present problem exists, and vice vert^a; while if the rising and setting limits form two distinct cun'Ors we have both a northern and southern limiting curve ; and the latter must evidently connect the extreme northern and southern points respectively of the two enclosed rising and setting curves. In our example of the eclipse of July 18, 1860, there exists only the southern limiting curve of the present problem, the penum- bral shadow passing over and beyond the north pole of the earth. Having assumed a value of (?, we find ^^ by the equations (502), (504) and (505), and then ^ by (509). This computed value of ^ and the assumed value of Q being substituted in (530), this equa- tion will be satisfied only when the true value of Q has been assumed. To find the correction of §, let us suppose that when the equation has been computed logarithmically we find log C/sin ((2 — i^) — log [a' + e sin {Q - Ey] =% If then dQ and rfj are the corrections which Q and ^ require in NORTHERN AND SOUTHERN LIMITS. 488 order to reduce x to zero, we have, by differentiating this equation, L a' +e sin (Q — E) J dO , dZ + -rz= — ^ + eBin(,Q — E)J A A: in which A is the reciprocal of the moduhis of common logarithms. In this differential equation we may neglect a' without sensibly affecting the rate of approximation. If then we put ^ dC ^ ZdQ we shall have dQ= ^"^ cot (§ — JB?) — cot (Q-^F) + g This value of dQ is yet to be reduced to seconds by multiplying it by cosec 1" or 206266". To find ^, we may take, as a sufficiently exact expression for computing dQj dZ, ^^dQ and by differentiating (502) (omitting the factor /t)^, which will not sensibly affect g\ cos p sin r dp + sin p cos y dy = — I cos Q dQ cos p cos Y dp — sin p Any dy z=z I sin Q dQ whence, by eliminating dy, dp ^ IsiniQ^r) dQ co&p By (505) a sufficiently exact value of f ^ for our present pur- pose is Cj = cos p whence d:, . ^dp — * = — sm p -^ dQ dQ g = lsinp sec* p sin (Q — f) (532) Putting, finally, (? = cot (Q - JB)— cot (§ - F) = - — sin (E -- F) ^^ ^ v^ y Bin(§ — JB?)sin(Q — iP)"^ ^ 484 SOLAR ECLIPSES. we have aq = f^:«I^" (5a*) in which 5.67664 is the logarithm of ^ X 206265". When the true value of Q has thus been found, the corre- sponding latitude and longitude on the earth's surface are found as in Art. 298. 312. The preceding solution of this problem (which is com- monly regarded as one of the most intricate problems in the theory of eclipses) is very precise, and the successive approxi- mations converge rapidly to the final result. For practical pur- poses, however, an extremely precise determination of the limit- ing curves of the penumbra is of little importance, since no valuable observations are made near these limits. I shall, there- fore, now show how the process may be abridged witliout making p. ^3 any important sacrifice of accuracy. In the first place, it is to be obsened that great precision in the angle Q .c I is unnecessary. If LM^ Fig. 43, is ^y^l J the limiting cun^e wliicli is tangent at A to the shadow whose axis is at ^ (7, and if (^ is in error by the quan- tity A CA\ the point determined will be (nearly) A' instead of A, Xow, altliough yl' may be at some distance from ^1, it is evident that it will still be at a proportionally small distance from the limiting curve. In fact, we may admit an error of sevvnil minutes in the value of Q without scnsibb/ removing the <-omputiMl point from the curve. The equation (530), which determines Q, may, therefore, without practical error be written under the approximate form esm{Q-E)=:jsmQ and in this we may employ for ^i the value Ci = cos /? Hence, having found fl from (502) by employing the first assumed value of Qj we then have sin (Q — E) / cos ^ NORTHERN AND SOUTHERN LIMITS. 485 whence tan (C — i^) = ^JtZ^^ tan i^ e — /cos p by which a second and more correct value of Q can be found. This equation will be readily computed under the following form : tan / = — cos ^5 ) « y (535) tan (C — \E) = tan (45^ + v') tan iE ) The value of Q thus determined may be regarded as final, and we may then proceed to compute the latitude and longitude by the equations (502) to (508). In this approximate method, loga- rithms of four decimal places will be found quite sufficient. 313. For the computation of a series of points by the preceding method, it is necessary first to determine the extreme times between which the solution is possible. It is evident that the first and last points of the curve are those for which ^j= 0, and, consequently, Q = Ej or Q = 180° + E. It is easily seen that these points are also the first and last points of the curve of maximum in the horizon (Art. 309), and, therefore, the limiting times are here the same as for that curve. If, however, we wish to determine these limiting times independently (that is, when the rising and setting limits have not been previously computed), the following approximative process will give them with all the precision necessary. Since Q = Ej or = 180® + E^ we have, at the required time, S = x^lBinE I .^3^. rj = y ^^ I cosE ) together with the condition (514), for which we may here employ e» + iy» = 1 If we put f = sin fy this condition gives jy = cos f. We have, by (512), ^ ^ em E = — COB E:= — e e and we may here regard e as constant. Let the required time be denoted by T= T^ + r, T^ being an assumed time near the middle of the eclipse. Let 6^,', e/, be the values of 6' and c' for 486 BOLAB ECLIPSES. the time T^, and denote their hourly changes by 6" and c" ; then we have, for the time Ty and hence, E^ being the tabulated value of E for the time T^ sin ^ = sin ^- H r cos E = cos E^A r ^ e ^ e If, also, Xq, y^y are the values of x and y for the time 7^ a/ and y' their hourly changes, we have x = x^ + x't y = y^J^tfx and the equations (536) become sin ;^ = x^ :p i sin ^^ -f I j/ qi — 6" jr cosr = yo^ ^cos^o + (y'=P — ^'1^ Let m, J!f, n, iV, be determined by the equations m sin Jlf = a:^j :+: ? sin E^ m cos Mz=zy^zf I cos -£?<> tt sin iV = a/ T — ft" ^ (537) n cos iV = y q= — c" in which the upper sign is to be used for the southern and the lower sign for the northern limit ; then, from the equations sin y = m sin M -\- n sin N. r cos z' = m cos M -{- n cos N . r we derive sin (/ — N) = m sin (3f — I^T) cos (z' — JV) = m cos ( Jf — iV) + nr Hence, putting y — iV= t^/, sin 4/ = m sin {M — iV) ^ ^ cos 4. m cos (M — i\r) . It is evident that cos i// is to be taken with the negative sign for the first point and with the positive sign for the last point of the curve. NORTHERN AND SOUTHERN LIMITS. 487 To find the latitude and longitude of the extreme points, we take Y =N+ '})/, tan y' = p^ tan y, and proceed by (519). Example. — To find the southern limit of the eclipse of Jul} 18, 1860. First. To find the extreme times. — Taking T^ = 2*, we have, from our tables, pp. 454, 455, and pp. 464, 466, Xo = — 0.0812 y^ = + 0.5961 I = 0.5367 E^ = 140 17' los: e = 9.7977 xf= + 0.5452 y' = — 0.1610 h" = + 0.0514 c" = _ 0.0151 where we take mean values of x', 3/', &c. From these we find by (537), taking the upper signs in the formube, log m = 9.3555 log n = 9.7182 Hence, by (538), log sin ( Jf — iV^) = n8.7354 log sin 4 = n8.0909 log cos 4= 0.0000 M = 2890 35' i\r=106 28 Jf— -^=183 7 log cos {M—'N) = n9.9994 m cos ( Jlf — N) n = + 0*.433 ^^=T 1.918 n T= — 1.480 or T = + 2 .346 Therefore, for the first and last points of the curve we have, respectively, the times Tj = 2* — 1».480 = 0».520 T, = 2 +2.846 = 4.346 To find the latitude and longitude of the extreme points corre- sponding to these times, we have log tan Y log p^ = 9.9987 log tan / Tint Point. LMt Point. 180» 42' — 0° 42' 287 10 105 46 n0.5102 nO.5492 n0.5089 nO.5479 21» 1'.4 20° 59'.8 6 19.2 63 42.7 488 Henoe, by (519), SOLAR ECLIPSES. w 102*> 40' 339*> 3(K 9 16 5 — 14 47 Second. To find a series of points on the curve. — ^We begin by computing the limits of Q for the hours 0*, 1*, 2*, 3*, 4% 5*. Thus, for 0* we have, from the table p. 465, and by (531), T log/ lege log tan V V log tan (45® + v) log tan } E log tan 4 0* 9.3882 9.8019 9.5863 2^ 5'.6 2 16.7 0.3533 8.5997 8.9530 5® 7'.7 7 24.4 For the southern limiting curve, § falls between E and \E-\-^^ i.e.j for 0*, between 4° 33' and 7° 24'. In the same manner we form the other numbers of the following table : T Lower limit of Q. Upper limit of Q. 0* 4° 33' 70 24' 1 9 22 15 18 2 14 17 23 13 3 19 14 30 53 4 24 8 38 4 5 28 55 44 36 The points of the curve are to be computed for times between 0*.520 and 4*.346, and we shall, therefore, assume for T the series 0*.6, 0*.8, 1*.0 4*.0, 4*. 2, which, with the extreme points above computed, will embrace the whole cun^e. Instead of determining Q for each of these times by the method of Art. 312, it will be sufficient to determine it for the hours 1*, 2*, 3*, 4*, and, hence, to infer its values for the inter- vening times. ThiL^, for T- 1*, assuming Q — 12®, which is • NORTHERN AND SOUTHERN LIMITS. 489 mean between its two limiting values, we proceed by the equa- tions (502), for which we can here use sin ^ sin /* = X — I sin Q sin ^ COS/* = y — I cos Q as follows : For T^=l\ ( X — 0.6266 log cos p 9.7396 V + 0.9170 0.5368 log{ 9.5923 Assume Q 12« log tan )/ 9.3319 a — X — ^ sin Q 0.7382 v' 12° 7'.1 b = y — I cos Q + 0.3920 \B 4 41.2 log a — log sin fi sin y n9.8682 log tan(45'»+ »') 0.1894 log b log sin fi cos y 9.5933 tan \E 8.9137 log sin fi 9.9221 tan(C — J-«^) 9.1031 C J-B 7" 13'.5 C 11 54.7 We thus find, for T= 1* 2* 8» 4* Q = IP 55', 22^ 20', 30" 16', 32" 17'. From these numbers we obtain by simple interpolation suffi- ciently exact values of Q for our whole series of points. And since it is plain from Art. 312, that even an error of half a degree in Q will not remove the computed point from the true curve by any important amount, we may be content to employ the following series of values as final : T Q T Q T C T Q 0».6 8» 1».6 18» 2».6 28» 3».6 31" 0.8 10 1.8 20 2.8 29 3.8 32 1.0 12 2.0 22 3.0 30 4.0 32 1.2 14 2.2 24 3.2 30 4.2 32 .5 I 1.4 16 2.4 26 3.4 31 For each time T we now take x, y, and l^ from the tables of the eclipse, and, with the value of Q for the same time, deter- mine the required point on the outline of the shadow by the 490 SOLAR S0LIPSE6. complete equations (502) to (508) inclusive, the use of which has already been exemplified in Art 302. Employing only four decimal places in the logarithms, we shall find that the carve may be traced through the points given in the following table : SOLAR ECLIPSE, July 18, I860.— SOUTHERN LIMIT. Gre«nwioh Mean Time. Latitude. liong. W. ttom Greenwieh. o 0».520 + 16° 5' 102" 40* 0.6 21 82 88 81 0.8 25 6 76 87 1.0 26 36 69 2 1.2 27 17 63 9 1.4 27 27 58 14 1.6 27 15 53 57 1.8 26 47 50 9 2.0 26 4 46 43 2.2 25 9 43 33 2.4 24 3 40 34 2.6 22 48 37 45 2.8 21 5 34 33 3.0 19 9 81 25 3 .2 16 41 27 50 3.4 14 14 24 39 3.6 11 9 20 44 3 .8 8 5 16 55 4.0 + 43 11 46 4.2 39 5 17 4.346 — 14 47 839 80 314. We have applied the preceding method only to the deter- mination of the extreme limits of the penumbra, which may be designated as the extreme limits of partial eclipse. The same method will determine the northern and southern limits of total or annular eclipse, by employing the value of / for the total shadow — that is, for interior contacts. The latter are, indeed, more important, practically, than tlie former, and therefore in CURVE OF CENTRAL ECLIPSE. 491 special cases somewhat greater precision might be desired than has been observed in the preceding example. In any such case, recourse may be had to the rigorous method of Art. 311. Since the limits of total or annular eclipse often include but a very narrow belt of the earth's surface, extending nearly equal distances north and south of the curve of central eclipse, they may be derived, with sufficient accuracy for most purposes, from this curve, by a method which will be given in Art. 320. The curve upon which any given degree of obscuration can be observed may also be computed by the preceding method. It is only necessary to substitute J for l, and to give J a value cor- responding to D according to the equation (529). All the curves thus found begin and end upon the curve of maximum in the horizon. Curve of Central Eclipse. 815. To find the curve of central eclipse upon the surface of the earth. — This curve contains all those points of the surface of the earth through which the axis of the cone of shadow passes. The problem becomes the same as that of Art. 298 upon the suppo- sition that the shadow is reduced to a point — ^that is, when Z — 1^ = 0, and, consequently, by (493), S = x 7j =y Hence, putting the equations (502) to (508) are reduced to the following ex- tremely simple ones, which are rigorously exact: sin p miy = x sin p cos r = 1/1 c sin C = y^ c cos = cos p cos f J sin i» = a: ) (539) cos f J cos 1* = c cos ((7 + ^1) sin f 1 = c sin (C7 -f- ^1) tan 0. - tan = ^-i— ^ = /J^t — V l/(l - ee) It will be convenient to prepare the values of y^ for the prin- cipal hours of the eclipse ; and then for any given time T taking the values of x, i/^ d^, /i^ from the eclipse tables, these equations determine a point of the curve. 492 SOLAR ECLIPSES. 316. The extreme times between which the solution is possible, or the beginning and end of central eclipse upon the earth, are found as follows. At these instants the axis of the shadow is tangent to the earth's surface, and the central eclipse is observed at sunrise and sunset respectively. Hence, Z being the zenith distance of the point Zy we have cos Z=Oy or, by (508), ^j = 0, whence, by (499), or which is equivalent to putting sin )9 = 1, or cos )9 = 0, in the first two equations of (539), so that we have sin Y =^ Xy cos r = yi Let a:' and y/ denote the mean hourly changes of z and y^ com- puted by the method of Art. 296. Let the required time of beginning or ending be denoted by T= Tf^+ r, T^ being an arbitrarily assumed epoch ; then, if (x) and (yj) are the values of X and 2/1 taken for the time TJ,? we have for the time T, sin Y = (.x") + x't cos r = 0/0 + y/r Let m, M, w, iV, be determined by the equations m sin M = (x) n sin N = sf m COS M = (yj) n cos iV = y/ then, from the equations sin Y = VI sm M -\- n sin iV. r cos Y = VI cos M -]- n cos iV. t we deduce, in the usual manner, sin (y — N) ^ VI sin {M — JV) cos (j — iV) = m cos (3/ — N) -\- nr or, putting 4' = r "" ^> ^^^ solution is sin 4 = m sin (M — -AT) cos 4. VI cos (M — N) n n I (540) (541) CURVE OF CENTRAL ECLIPSE. 493 where cos i^ is to be taken with the negative sign for the beginning and with the positive sign for the end. To find the latitude and longitude of the extreme points cor- responding to these times, we have, in (539), cos )9 = 0, sin )9 = 1, and, therefore, C= 90°, c = cos y : hence, taking y = iV+ '^j COS 9>j sin ^ = sin /* cos 9>j cos 1^ = — cos Y sin d^ sin 9>j = cos Y cos d^ \ (542) tan w. tan cp = ^2 — w = fjL. — * V^Cl - ee) '^ 317. To find the duration of total or annular eclipse at any point of the curve of central eclipse. — This is readily obtained from numbers which occur in the previous computations. Let r=the time of central eclipse, t = the duration of total or annular eclipse, then T'= T^ J< is the time of beginning or end. Let x and y be the moon's co-ordinates for the time T; f and rj those of the point on the earth at this time ; x\ y\ f ', jy', the hourly in- crements of these quantities ; then, at the time T' we have, by (491), (I - iC) sin C = x =F la/^ — (f =p K'O (I — iZ) COB Q = y qp iy't — (i^ T } 1^7) But we here have a: = f , y = 3^, and we may put ^ = ^^ = cos )9, whence (I ^ i cos fi) sin C = =F (^ — f — (I — I cos /5) cos C = =F (y' — 7i') -- For the values of f and jy' we have, with sufficient precision, since t is very small, $' =: fi ( — y &in d + cos ^5 cos d) 1^'= fj! X sin d Hence, by (511) and (512), we find, very nearly, a/ — - f ' = </ — f/Qos^ dcos fi = c^ — fcosfi y'~V = -y If, therefore, we put L^l — icoBtS a z:^ (f^f COB fi (648) 494 SOLAS SCLIPSBS. we have X8inO = — Xco6Q = — 2 2 where we omit the doable sign, since it is only the numerical value of t that is required. Hence, we have, for finding tj the equations , ^ a ^ 7200i8inQ ,,,,. tan Q = — t = ^ (;M) the last equation being multiplied by 3600, so that it now gives t in seconds. The value of cos j9 is to be taken from the computation of the central curve for the ^ven time T, and i, log i, log/, c', V, from our eclipse tables. 318. To find where the central ectipse occurs at noon. — In this case we have, evidently, x = 0, and hence, in (539), sin fi = y, (545) by which ^ is to be found from the value of yj which corresponds to the time when x = 0. We then have C= ^, c = 1, tf = 0, and therefore the required point is found by the formulae 9i=fi + d, « = M, (546) in which d^ and /i^ are taken for the time when r = 0. 819. The formulffi (539), (545), and (546) are not only extremely simple, but also entirely rigorous, and have this advantage over the methods commonly given, that they require no repetition to take into account the true figure of the earth. It may be observed here that the accurate computation of the central curve is of far greater practical importance than that of the limiting curves before treated of. The formulje (541) must be computed twice if we wish to obtain the times of beginning and end with the greatest pos- sible precision ; for, these times being unknown, we shall have at first to employ the values of x' and y' for the middle of the eclipse, and then to take their values for the times obtained by the first computation of the formulae. With these new values a second computation will give the exact times. CURVE OF CENTRAL ECLIPSE. 495 Example. — To compute the curve of central and total eclipse in the eclipse of July 18, 1860. It is convenient first to prepare the values of yi = ~ for the principal hours of the eclipse, as well as its mean hourly differ- ences. With the value log p^= 9.99873 we form, from the values of y given in the table p. 464, the following table : Gr. T. yi y/ 0» + 0.91972 — 0.16095 1 .75896 114 2 .59782 132 3 .43638 149 4 .27450 166 5 .11237 182 To find the times of beginning and end we may assume 7^= 2* ; and for this time we have (x ) == m sin Jf = — 0.08124 a/— n sin i\r— + 0.5453 (y^) — m cos Jlf — + 0.59782 y/ — n cos iV^ — — 0.1613 whence log m — 9.78054 log n = 9.7548 M= 352oi5'40" N— 106^28'.7 Employing but four decimal places in the logarithms for a first approximation, we find, by (541), mco8(J[f — IT) n cos 4^ n = + 0*.435 = =F 1 .468 '"i — 1 .033 '"a - + 1 .903 Approximate time of beginning = 2* — 1*.033 = 0» .967 « n end = 2 + 1 .903 . = 3 .903 Taking now x^ and y/ for these times respectively, and re- peating the computation^ we have 496 80LAR ECLIPSES. y^ = n COS N logn N m cos (M — N) n cos 4 n T 4 Beginning. End. + 0.54531 + 0.54525 — 0.16113 — 0.16164 9.75482 9.75489 106° 27' 42" 106° 30* 45" + 0».4349 + 0».4357 — 1 .4684 + 1 .4685 0.9665 3.9042 213° 23' 12" 326° 37' 40" For the latitude and longitude of the points of beginning and end, we now take t' = iV + '\//> and with the values of d^ and /ij (pp. 455, 464) for the above computed times, we have r = ^+4 whence, by (542), 9 w Local App. Time = ^ Beginning. 319° 50' 54" 21 1 15 13 1 1 End. 73° 8' 25" 21 57 5 3 45° 36' 50" 126 3 8 16* 27"'.9 15° 45' 34" 320 53 9 6* 24".8 For the series of points on the curve we take the times 1*.0, 1'^.2, 1*.4 8*.6, 8*.8, which are embraced within the extreme times above found, and proceed by (539). Thus, for 2'^.0 we have T X = sin /9 sin y y^ = sin p cos y log sin /9 log cos /9 = log c cos C log y^ = log c sin C logc C 2*. — 0.08124 + 0.59782 9.78054 9.90173 9.77657 9.99856 36° 51' 21" 21 49 CURVE OF CBNTRAL ECLIPSE. 497 logo: log coos ((7 + ^,) log c sin ((7 + dj log COS f J sin «9 log cos f , COS ^ log sin f>, App. Time = * in time, 57^ 52' 10" n8.90977 9.72435 9.92636 351« 17' 13" 28 31 12 37 13 59 57 39 20 23* 25- 8'.8 For the duration of totality at this point, we take from pp. 454, 464,465, I = — 0.009082 h'= + 0.1532 log I = 7.6608 c' = + 0.6011 log/= 9.3883 and hence, with log cos ^ = 9.9017 above found, we obtain, by (543), i = — 0.012734 « = + 0.4061 and, by (644), disregarding the negative sign of i, t = 211*.3 = 3- 31*.3 For the place where the central eclipse occurs at noon, we find that x == at the time T= 2*. 149, at which time we have y,_8my9 + 0.57878 ? 85» VbZ" d. 21 45 9i 56 1 38 9 56 6 57 IL=:-IO 80 45 18 The whole curve may be traced through the points given in the following table : Vol. 1.-81 498 SOLAR BCLIPSE8. SOLAR ECLIPSE, July 18, I860.— CURVE OF CENTRAL AND TOTAL ECLIPSE. Greenwich Latitude. Long. W. Arom Greenwich. App. Local Time. Duration of Mean Time. * U ^ Totality. 0».967 45" 86'.4 126' S'.l 16* 27-.9 1.0 60 37.8 113 11.6 17 21 .3 2- ' 1'.5 1.2 57 16.2 89 14.6 19 9 .1 2 35.1 1.4 69 29.1 72 52.8 20 26 .6 2 55.8 1.6 59 55.1 69 6.2 21 33 .7 3 11.4 1.8 59 11.6 47 16.6 22 33 .0 3 23.1 2.0 57 39.3 37 14.0 23 25 .1 3 31.8 2.149 56 7.0 30 45.3 .0 3 34.7 2.2 66 31.5 28 42.6 11 .2 3 36.2 2.4 62 56.9 21 26.1 52 .4 3 38.0 2.6 50 0.9 15 3.9 1 29 .8 3 36.4 2.8 46 46.3 9 21.8 2 4 .6 3 32.0 8.0 43 13.6 4 2.2 2 37 .9 3 24.6 3.2 39 20.7 358 47.1 3 10 .9 3 14.4 3.4 35 1.6 363 12.5 3 45 .3 3 1.1 3.6 30 1.5 346 35.4 4 23 .7 2 43.5 3.8 23 28.5 336 44.1 5 15 .1 2 18.5 3.904 15 45.6 320 53.2 6 24 .8 Northern and Southern Limits of Total or Annular Eclipse, 820. To find the northern and southern limits of total or annular eclipse. — As already remarked in Art. 314, these limits may be rigorously determined by the method of Art. 311, by taking I = the radius of the umbra {i.e. for interior contacts) ; but I here propose to deduce them from the previously computed curve of central eclipse. This radius I is assumed to be so small that we may neglect its square, which can seldom exceed .0003, and this degree of approximation will in the greater number of cases suffice to determine points on the limits witliin 2' or 8', which is practically quite accurate enough. The two limiting curves of total or annular eclipse, then, lie so near to the central curve that the value (^^ = cos ^, for a given time r, already found in the computation of the latter curve, may be used for the former in the approximate equation which determines Q. We can, therefore, immediately find Qhj (535), — i.e. LIMITS OF TOTAL OR ANNULAR ECLIPSE. 499 tan v' = — COS p ) e V (547) tan (C — \E) = tan (45^ + /) tan }^ ) where/, e, and -Bare to be taken from the eclipse tables for the lime T. The co-ordinates of the point on the central curve correspond- ing to the time T being S = x and y^ = 7jy (Art. 315), those for a point on the limiting curve may be denoted hy x + dx and Vi + ^!/v These being substituted for f and tj^ in the equations (499), we have dx = — (I — iCj) sin Q ^^i = — G — ^^i) cos Q where in the expression for dy^ we omit the divisor /o^, as not appreciably changing the value of so small a term. Let ^1, t?, 10 be taken from the computation of the central curve for the time T, and let f^ + d<p^^ lo + do), be the cor- responding values of f^ and (o for the point on the limit for the same time. Then, by difterentiating (500), observing that rftf = — dof, we have cos f J cos ^ rf a* -f- si'^ 9t sin i> d^^ = — dx cos f J sin ^ d(o — sin f ^ cos ^ d^^ = — dy^ sin d^ + dZ^ cos d^ cos ^^d^^ = dy^ cos d^ -\- dZ^ sin d^ whence, by eliminating rff j and substituting f ^ for its value given by the third equation of (497), we find Ct cos f^dat = — dx (cos f>, cos dj -}- sin ^^ cos ^ sin rfj — rf^j sin f ^ sin d Cjdf>,= — dx sin 1* sin <fj -|- rfy, cos ti> Hence, substituting cos ^ for f j, dot = ^ (cos ^ sin § sin rfj + sin d^ cosQ) tan ^ cos^ , ? — I cos ^ . ^ , _| L_ 8in § cos d^ COS/5 Z "^^ t cos B df = (sin ^ sin § sin d^ — cos * cos Q) COS/? 304 SOLAR ECLIPSSS. These values are yet to be divided by sin 1' to reduce them to minutes of arc. It will be convenient to put r=— i- t'= sin 1' an V ^^ l^icoQp _ V ., (548) cos /9 Bin r cos/9 in which Z', i', and >l will be expressed in minutes. We may in practice substitute df for dy ^ within the limits of accuracy we have adopted ; for we find, from the equations on p. 457, J dtp, cos* w ,1 — ee sin* w dip •=. * . = o^pj |/(1 — ee) cos'fj |/(1 — ee) where the multiplier of dtp^ cannot differ more from unity than l/(l — ee) does, — t.e. not more than 0.00835: so that the substitu- tion of one for the other ^vill never produce an error of 1' so long as dtp^ is less than 5^. Finally, adapting the values of doi and dip for logarithmic computation, by putting A sin -H" r= cos Q A cos IT = sin Q sin d^ we have ^ (549) dm =iX\]i cos (i? — H) tan ip^ + sin Q cos dj d^ = Xh sin (t> — H) The forraulte (547) give two values of Q diflfering 180°. The second value will evidently give the same numerical values of dw and d<py but with opposite signs ; and therefore we may com- pute the equations (549) with only the acute value of Q^ and then the longitude and latitude of a point on one of the limits are at -f- dio, ip -\- df and those of a point on the other limit are w — duty <p — dtp The first of these limits will be the northern in the case of total eclipse, but the southern in the case of annular eclipse, observing always to take I with the negative sign for total eclipse, as it comes out by the formulae (487) and (489). LIMITS OF TOTAL OR ANNULAR ECLIPSE. 601 It is evident that this approximate method is not accurate when cos j9 is very small, that is, near the extreme points of the curves; and it fails wholly for these points themselves, since cos j9 is then zero and the value of X becomes infinite. These extreme points, however, are determined directly in a very simple manner by the formulse (536), (537), (538), combined with (619), by employing in (536) and (537) the value of I for interior contacts; and it is with these formulae, therefore, that the com- putation of the limits of total or annular eclipse should be com- menced. Example. — ^Find the northern and southern limits of total eclipse in the eclipse of July 18, 1860. I\rsL To find the extreme points. — The values of 6' and c' for exterior contacts, from which the values of E on p. 465 are derived, differ so little from those for interior contacts that in practice, unless extreme precision is required, we may dispense with the computation of the latter. For our present example, therefore, taking the value of -Efor 7^= 2* and the mean value of log 6, as in the computation of the extreme points of the southern limit for the penumbra, p. 487, together with I — 0.0091 we find, by (537), for the northern limit, log m = 9.7854 M — 352*' 33'.6 log n = 9.7553 J\r = 106 2r.o and for the southern limit, log m — 9.7731 Jf=85P 55'.0 log n = 9.7542 N — 106 27.0 Hence, by (538), Northern Limit. Sonthem Limit. 4 T First Point. 213<> 54'.3 0*.976 Last Point. 326^ 5'.7 3*.892 First Point. 212^ 39^.0 0».951 Last Point. 327^ 2r.O 3».917 Taking ;• = N+ a//, and the values of d^ and fi^ for these times respectively, with log Pi = 9.9987, we find, by (518) and (619), 502 log tan / SOLAR ECLIPSES. r 320^ 21'.3 72° 32'.7 319° 6'.0 73° 48 .0 / JI9.9170 0.6012 n9.9363 0.6356 d. 21° 1'.2 21° O'.O 21° V2 21° (K.O « 246 31.7 96 26.7 247 26.7 96 67.7 ^ 13 9.6 66 64.1 12 47.1 67 16.6 m 126 37.9 320 27.4 126 20.4 821 18.9 9 46 7.7 16 21.6 46 2.8 16 11.4 Second, To find a series of points between these extremes, by the aid of the curve of central eclipse, we assume the same series of times as in the computation of that curve, and proceed by (547), (548), and (549) ; to illustrate the use of which I add the computation for r= 2* in full. From the computation, p. 496, we have log cos p log tan ^^ d. For r= 2* (O 9 9.9017 0.1970 361° 17'.2 21 0.8 37 14.0 67 39.3 Then, by (547), / (p. 465) log log 008 p log tan / \E log tan (45° + »/) log tan 1 E logtan(§— 1J5?) Q-\E Q Hence, by (549), 9.5953 9.9017 9.4970 17° 26'.0 7 8.7 0.2823 9.0982 9.3805 13° 30'.3 20 39.0 By (548), I log I log I' log i log V V V sec p X — 0.009082 n7.9582 nl.4946 7.6608 1.1971 16'.74 — 39 .16 — 64.90 log cos Q == log h sin H log sin Q sin d^ = log h cos H log h H 9.9712 9.1020 9.9761 82° 18'.2 268 69.0 LIMITS OF TOTAL OR ANNULAR ECLIPSE. 503 log A log A log COB (^ — If) log tan f>| log (1) log A log sinQ cos <f, log (2) (1) (2) dw nl.7396 9.9751 n8.2490 0.1970 0.1607 nl.7396 9.5175 nl.2571 + 1'.45 — 18 .08 - 16 .63 log^ log A log sin (d—^ log d^ d<p nl.7396 9.9751 n9.9999 1.7146 + 5r.83 Hence, for the time r= 2*, we have the two points, to 9 dot dtp N. Limit. 36^ 57.'4 58 31.1 S. Limit. 37^ 30'.6 56 47.5 SOLAR ECLIPSE, July 18, 1860. Northtm Limit of Total EeUpte. Southern Limit of Total Eel^^se, Latitude. • Longitude. Gr. T. ^ U 0».976 46^ 8' 126*> 38' 1.0 50 18 116 27 1.2 57 47 90 57 1.4 60 13 74 1.6 60 46 59 40 1.8 60 4 47 23 2.0 58 31 36 57 2.2 56 21 28 9 2.4 53 43 20 40 2.6 50 43 14 12 2.8 47 24 8 44 3.0 43 47 3 1 3.2 39 49 357 43 3.4 35 25 352 6 3.6 30 18 345 23 3.8 23 31 335 8 3.892 16 22 320 27 Latitude. Longitude. Or. T. ^ u 0*.951 45^ 3' 125^ 20^ 1.0 50 57 109 56 1.2 56 45 87 33 1.4 58 45 71 46 1.6 59 4 58 31 1.8 58 19 47 11 2.0 56 48 37 31 2.2 54 42 29 16 2.4 52 11 22 10 2.6 49 19 15 56 2.8 46 9 10 39 3.0 42 41 5 3 3.2 38 52 359 51 3.4 34 38 354 20 3 .6 29 45 347 48 3.8 23 26 338 20 3.917 15 11 321 19 SOLAR ECLIPSES. 321. The curves above computed j lowing chart. i all exhibited in the fol- Tor the construction of sueh charts, on even a much lai^er eealc, the degree of aeeuracy with which our computations have been made \a far greater tlian ia necessary, and many abridgments may he made which will readily occur to the skilful computer.* r a grnphic melhod of cam true ting eclipse chnrle, mi k paper bf Hf. OBT, ProcmdingB of tha Am. Asaaoiutian for Ihn Adr. of ScieiiM, 8lb PREDICTION FOR A GIVEN PLACE. 505 Prediction of a Solar Eclipse for a Given Place. 322. To compute the time of the occurrence of a given phase of a solar eclipse for a given place. — The given phase is expressed by a given value of J, and we are to find the time when this value and the co-ordinates of the given place satisfy the conditions (485). This can only be done by successive approximations. Let it be proposed to find the time of beginning or ending of the eclipse at the place. The phase is then J = i — i^, and we must satisfy the equations (491). Let T^ be an assumed time, and r= Tq-{- r the required time. Let a:, y, x', y\ rf, ?, log i, be taken from the eclipse tables (p. 454) for the time T^, Assuming that X and y vary uniformly, their values at the time T are X + x'r and y + y't. The co-ordinates of the place at the time Tq are found by (483) or (483*), in which fx is the sidereal time at the place. Putting »> = /* — a ^ fi^ — at in which at is the west longitude of the place and /i^ may be taken from the table (p. 455) for the time T^, the formulse become ABmB = pBin^ S = P cos y»' sin * "j A cos B = p cos f>' cos * Tj =A sin (B — d) > (550) C=^cos(^ — rf) j Let f, ^' denote the hourly increments of f Bmdrj; then, assuming that these increments also are uniform, the values of the co-ordi- nates at the time T are f + f '^ and tj + yfr. The values of f ' and yf are found by the formulae (p. 462) f = i^ p cos f' cos ^ V=A*'f sin d — d'ti in which y! and d' are the hourly changes of /i and d multiplied by sin 1". The rate of approximation will not be sensibly affected by omitting the small term cf'^, and the formulae for f ' and yf may then be written as follows : r = A*' -A cos * ^ = A*'f sm d (551) Put i = ? — i: then, neglecting the variation of this quantity in the first ap- proximation, the conditions (491) become, for liie time T, i sin § = :r — e + (a/ — e') T i cos C = y — iy + (y — ly') T 506 80LAR ECLIPSES. Let the auxiliaries m, -Bf, n, and iVbe determined by the eqna- tiona mcoQM=y — i) neo8JV=y' — if )^ ^ then, from the equations iy sin Q = m sin M -\- n sin JV. r Z cos Q = m cos ilf -|- n cos JV. r we deduce i sin (§ — JVO = m sin {M-^N) L cos (Q — N) = m cos (Jlf — -AT) -(- nr Hence, putting oj' = § — i^T, wo have m sin ( Jtf" — If) Bin 4 = ^^ ' L L cos 4 m cos {M — N) n n (558) by which r is found. Since the first of these equations does not determine the sign of cos '^, the latter may be taken with either the positive or the negative sign. We thus obtain two values of T^= Tq + 7> the first given by the negative sign of fi being the time of beginning, and the second given by the posi- tive sign being the time of ending of the eclipse at the place. For a second approximation, let each of the computed times (or two times nearly equal to them) be taken as the assumed time Tq^ and compute the equations (550), (551), (552), (553) for beginning and end separately. The first approximation may be in error several minutes, but the second will always be correct >\'ithin a few seconds, and, therefore, quite as accurate as can be required ; for a perfect prediction cannot be attained in the present state of the Ephe- merides. The formula for r may also be expressed as follows : m sin {M — N — 4*) n sin 4 which in the second approximation will be more convenient than the former expression ; but when sin n^ is very small it will not be so precise. PREDICTION FOR A GIVEN PLACE. 507 If we put t = the local mean time of beginning or end, we have 323. The prediction for a given place being made for the purpose of preparing to observe the eclipse, it is necessary also to know the point of the sun's limb at which the first contact is to take place, in order to direct the attention to that point. This is given at once by the value of which is the angular distance of the point of contact reckoned from the north point of the sun's limb towards the east (Art. 295). The simplest method of distinguishing the point of contact oii the sun's limb is (as Bessel suggested) by a thread in the eye-piece of the telescope, ari'anged so that it can be revolved and made tangent to the sun's limb at the point. The observer then, by a slow motion of the instrument, keeps the limb very nearly in contact with the thread until the eclipse begins. The position of the thread is indicated by a small graduated circle on the rim of the eye-piece, as in the common position micrometer. This method is applicable whatever may be the kind of mounting of the telescope. Nevertheless, if the instrument is arranged with motion in altitude and azimuth, it will be conve- nient to know the angle of the point of contact from the vertex of the sun's limb, which is that point of the limb which is nearest to the zenith. The distance of the vertex from the north point of the limb is equal to the parallactic angle which being here denoted by y^ is found, according to Art. 15, by the formulae p sin y z= cos ^ sin i^ p cos z' = sin y» cos d — cos f sin d cos ^ (where we have put p for sin !^ and & for the sun's hour angle). As Y is not required with very great accuracy, we may here take [see (494)] pBiny^S p cos r = V in which f and rj are the values of the co-ordinates of the place at the instant of contact. But, if f and rj denote the values at the time T^ we must take p Biny = ? + S't pcosr = t^ + 1?'^ (554) 508 SOLAR ECLIPSES. Ill which we employ the values of f, jy, f ', yf^ and r furnished by the last approximation. We then have Angular diBtance of the point of contact ftrom 1 = C — f T'Vfi'i^ the vertex towards the east, J=iV-}-4 — T 324. To find the instant of maximum obscuration for a given placCj and the degree of obscuration. — At the instant of greatest obscura- tion the distance J of the axis of the shadow from the place of observation is a minimum.* If we denote the required time by Ti = jTq + ^1) the equations of Art. 822 determine r^ for a given value of J if we substitute J for L. Denoting the value of § — iV for this case by '^i, we have, therefore, J sin 4i = m sin (M — N) A cos 4j^ = m cos ( Jtf" — N) + nr^ the sum of the squares of which gives J« = m« 8in« (Jf — JVO + [m cos (2f — iST) + nr,]' Since m and M are computed for the time 7J,, and N is sensibly constant, the term m' sin' (M — N) is constant, and therefore J is a minimum when the last term is zero, that is, when m cos {M — N) n (656) which quantity is already known from the computation of (553). We have, also, J == it m sin (^af — JV^ = ± i sin 4, (557) in which the sign is to be so taken as to make J positive. The degree of obscuration is then given by the formula (Art. 310), L + L, in which D is expressed in fractional parts of the sun's diameter, and L and L^ are the radii of the penumbra and umbra (the * More strictly, Z ~ J is a maximum, as in Art. 809 ; but we here >mh||{^^^^ small variation of L. The rigorous solution of the problem may be M^- •— ^"^^ the condition (526) P' = ; but the above approximation is sufficfen^ " PREDICTION FOR A GIVEN PLACE. 609 latter being negative) for the place of observation. From (488) we find, by putting 8ec/= 1, and hence L — A D = (658) in which k = 0.2723. If we neglect the augmentation of the moon's diameter, or, which is equivalent, the small difference between L and 2, and put V we have ' ^ ^ 2> == e qp e sin 4» where the lower sign is to be used when sin '^ is negative, so that D is always the numerical difference of e and e sin a//. Li this form e may be computed for the eclipse generally, and n^ will be derived from the computation for the penumbra for the given place. A preference should be given to the value of a// found from the computation for the time nearest to that of greatest obscuration, which is usually that used in the first approximation of Art. 322. Example. — ^Find the time of beginning and end, &c., of the eclipse of July 18, 1860, at Cambridge, Mass. The latitude and longitude are f> = 42° 22' 49" Id = 71° 7' 25" For this latitude we find, by the aid of Table III., or by the formulse (87), log p sin tp' = 9.82644 log p cos / = 9.86912 With the aid of the chart, p. 504, we estimate the time of the middle of the eclipse at Cambridge to be not far from 1*. Hence, taking 7J, = 1* for our first approximation, wo take for this time, from the eclipse tables, p. 454, z r= — 0.6266 ^=-\- 0.6468 I = 0.6868 y = -f 0. 7667 y* = — 0. 1606 log i = 7.66287 rf= 200 67'.4 ^1= 180 8r.2 log /*'= 9.41799 610 SOLAB SCLIPSES. Hence, by (550) and (551), fii—o — '» — 802O28'.8 B = 59 24.6 f- 0.6246 V — 4-0.4844 r= -1-0.1038 ^ = — 0.0686 and, by (552) and ( [553), mBin M z — ^ — 0.0020 m cos Jf y — 7 + 0.2728 logm 9.4850 M = 8590 84'.7 M—N — 256 84.1 log sin 4 n9.6966 log 008 4 -= 9.9887 log^= 9.7858 t; = 0.0028 /; = / — if= 0.5340 n sin ^'= z*— ^'=r + 0.4416 n cos iV= y'— tf = — 0.1020 log n = 9.G562 JV= 103«0'.0 m cos (Jf — K) n L cos 4. + 0*.140 _ 1.028 _ f — 0.888 ^~"lor+ 1.168 Approximate time of beginning = 0^.117 end = 2 .163 <( i< Taking then for a second approximation T^ = 0*.12 for begin- ning, and Tq= 2^.16 for end, we shall find* X y of d I log: Beginning. End. 0M2 2M6 1.10642 + 0.00601 + 0.89783 + 0.57034 4- 0.54528 + 0.54530 0.16015 — 0.16090 20° 57' 45" 20° 56' 53" 19 8 30 55 13 0.53686 0.53673 289° 11' 43" 319° 47' 48" 0.69868 — 0.47755 + 0.53915 + 0.42423 9.66935 9.88504 + 0.06368 + 0.14793 0.06544 0.04470 * The values of 7f and y' here employed are not those given in the table p. 446, but their actual values for the time T^^ as given in the table of zf and y' on p. 404. PREDICTION FOR A GIVBN PLACE. 511 iZ L m Bin M m Qo^M log m M n sin N n cos JV log n N 4 T .{ (O t { Beginning. End. 0.00215 0.00353 0.53471 0.53320 -- 0.40774 + 0.48356 + 0.35868 + 0.14611 9.73484 9.70342 311^ 20' 16" 73° 11' 15" + 0.48160 + 0.39737 -- 0.09471 — 0.11620 9.69093 9.61702 101^ r 32" 106*^ 18' 0" 210 12 44 326 53 15 210 44 328 49 56 — 31' 16" — 1° 56' 41" + 0*.0197 + 0*.0800 0M397 2 .2400 0* 8-23- 2* 14-» 24- 4 44 30 4 44 30 19 23 53 21 29 54 July 17. July 17. Sir 51' 32" 75*^ 7' 56" Local time, Angle of Pt.of Contact from \ North Pt. of the sun = V A third approximation, commencing with the last computed times, changes them by only a fraction of a second. To find the angular distance of the point of contact from the vertex of the sun's limb, we have from the second approximation, by (564) and (555), f + f 't = ;) sin r Tj -^ Tj'r =p cos ^ r Angle from vertex = Q — y Beginning. — 0.6974 + 0.5379 307° 38'.8 4 12.7 End. -- 0.4658 + 0.4206 312° 4'.5 123 3.4 The time of greatest obscumtion is best found from the first approximation, which gives, by (656), £12 SOLAR ECLIPSES. T = 1*. _mcoB(3f-J\r)^' ^^^^ n * ' Tj = 1M40 = 1* 8* 24'. w = 4 44 80 Local time of max. obscur. = t = 20 23 54 For the amount of greatest obscuration we have, also, from the first approximation, by (667) and (568), i = 0.6340 logi= 9.7275 k = 0.2723 log sin 4 == n9.6955 Z^k =0.2617 logJ=: 9.4230 2(i — A:) = 0.5234 J= 0.2649 i — J= 0.2691 0.5234 Or, by (559), taking as constant the value of e found by employ- ing the mean value I = 0.5367, i.e. e = 1.015 we have « gin 4 = — 0.503 D = 0.512 which is quite accurate enough. 326. Prediction for a givai place by the method of the American Uphemeris. — This method is based upon a transformation of Bessel's formula suggested by T. Henry Safford, Jr., and, with the aid of the extended tables in the Ephemeris, is somewhat more convenient than the preceding. The fundamental equa- tion (490) gives, by transposition, (x — ?)*=(/-: tan/)« — (y — 7))' the second member of which may be resolved into the factors 6=(Z_:tan/) + (y-i?) c = (Z — C tan/) — (y—ri) or, by (494), b = 1 -{- y — /> sin ^' (cos d -{- am d tan/) + p cos (p' (sin d — cos d tan /) cos ^ c = 1 — y -f- /> sin ^' (cos d — sin rf tan /) — P cos y' (sin d + cos d tan /) cos * PREDICTION FOR A GIVEN PLACE. 513 If we put A=x B = l + y C= — l + y E = coad + Bin d tan/ = cos (d — /) sec/ F = COB d — Bin d tan/= cob (d +/) sec/ G=i Bin d — cos d tan/ = sin (d — /) sec/ H=Bin d + cos d tan/ = sin (d + /) sec/ all of which are independent of the place of observation and are given in the Ephemeris for each solar eclipse, for successive times at the Washington meridian, we shall then have to com- pute for the place a = X — ( = A — p cos ^ sin ^ ^ 6 = -B— -^ /» sin ^+ G^ /» cos ^ cos * I (560) c = — G -\- F pBin^ — JEp cos ^' cos d ) and the fundamental equation becomes We have here, as before, t? = /i^ — a; ; and the value of fi^ is also given in the Ephemeris for the Washington meridian. If now for any assumed time T^ we take from the Ephemeris the values of these auxiliaries, and, after computing a, 6, and c by (560), find that a differs from >/6(?, the assumed time requires to be corrected ; and the correction is found by the following process. Put m = i/bc, o^, y, m'= the changes of a, b, m, in one second, r = the required correction of the assumed time ; then at the time of beginning or ending of the eclipse we must have a + o't = m -f m'r whence m — a a' — m' To find a' we have, by differentiating the value of a and de- noting the derivatives by accents, a'= A' -^fi'p cos ^ cos * (661) Vol. L— 83 514 SOLAB E0LIP8E& in which fJ denotes the change of //^ in one second, and is the same as the fJ of our former method divided by 3600. To find ml we have, following the same notation, and neglect- ing the small changes of E^ F^ G-j JS, 2, and/, V = B' ^ /i' Gp cos ^ sin d^ d = — C + plRp cos f>' sin * Since / is small, we may in these approximate expressions pot G = Hj and hence V= — d= B'— fi! Gp cos f^ sin^ (561*> Now, from the formula m* = 6c, we derive 2 mm' = d>'+ h(f= (c — 6)^ which, if we assume becomes m'= — ycotQ and therefore r is found by the formula m — a a' + 6' cot Q (563) The Ephemeris gives also the values of A', 5', and C, which are the changes of -4, J5, and C in one second. These changes being very small, the unit adopted in expressing them is .000001; so that the above value of r, as also the value of /£' in (561), must be multiplied by 10^ The formulse (560-568) then agree with those given in the explanation appended to the Ephemeris. It is easily seen that Q here denotes the same angle as in the preceding articles ; for we have at the instant of contact tan Q = = = m! h — c y — ly Examples of the application of this method are given in every volume of the American Ephemeris. CORRECTION FOR RETRACTION. 51b 826. The preceding articles embrace all that is important in relation to the prediction of solar eclipses. Since absolute rigor is not required in mere predictions, I have thus far said nothing of the effect of refraction, which, though extremely small, must be treated of before we proceed to the application of observed eclipses, where the greatest possible degree of precision is to be sought CORRECTION FOR ATMOSPHERIC REFRACTION IN ECLIPSES. 327. That the refraction varies for bodies at different distances from the earth has already been noticed in Art. 106 ; but the difference is so small that it is disregarded in all problems in which the absolute position of a single body is considered. Here, however, where two points at very different distances from the earth are observed in apparent contact, it is worth while to inquire how far the difference in question may affect our results. Let SMDA, Fig. 44, be the path of the ray of light from the sun's limb to the observer at A^ which touches the moon's limb at M; SMB the straight line which coincides with this path between 5^ and M^ but when produced intersects the vertical line of the observer in B, It is evident that the observer at A sees an ap- parent contact of the limbs at the instant when an observer at B would see a true contact if there were no refiraction. Hence, if we substitute the point B for the point A in the formula of the eclipse, we shall fully take into account the effect of refraction. For the purpose of determining the position of the point By whose distance from A is very small, it will suffice to regard the earth as a sphere with the radius p = CA, It is one of the pro- perties of the path of a ray of light in the atmosphere that the product qfx sin i is constant (Art. 108), q denoting the normal to any infinitesimal stratum of the atmosphere at the point in which the ray intersects the stratum, fx the index of refraction of that stratum, and i the angle which the ray makes with the normal. 516 60LAB ECLIPSES. If, then, /t>, [jL^y Z' denote the values of 3, //, and i for the point -4, we have, as in the equation (149), qix Bin I = p[i^ sin Z' in which Z' is the apparent zenith distance of the point M^ and /1q is the index of refraction of the air at the observer. Now, let us consider the normal q to be drawn to a point D of the ray where the refractive power of the air is zero, that is, to a point in the rectilinear portion of the path where /i = 1. Then our equation becomes q sin i = pix^ sin Z' in which q = CD, i = MDF = CDB. Putting Z = the true zenith distance oi M= MBVj and s = the height of B above the surface of the earth = ABy the triangle CDB gives (p + s) BinZ = q sin i which with the preceding equation gives - 5 Au sin Z' ,^, In order to substitute the point B for the point A in our com- putation of an eclipse, we have only to write p + s for p in the equations (483), or /> 1 1 H — J for />. Therefore, when we have computed the values of log f, log ij, and log !^ by those equa- tions in their present form, we shall merely have to correct them by adding to each the value of log ( 1 H I This logarithm may be computed by (564) for a mean value of /Ji^ (= 1.OOO2800) and for given values of Z. For Z we may take the true zenith distance of the point Z (Art. 289), determined by a and d. But by the last equation of (483) we have so nearly cos Z= ^ that in the table computed by (564) we may make log ^ the argu- ment, as in the following table, which I have deduced from that of Bessel {Astron. Untersuchungen, Vol. II. p. 240). SEDUCTION TO THE SEA LEVEL. 517 logC Correction of logs, of ^ J7, C- 0.0 9.9 9.8 9.7 9.6 0.0000000 .0000001 .0000002 .0000006 .0000008 9.6 9.4 9.3 9.2 9.1 0.0000014 .0000023 .0000036 .0000064 .0000081 9.0 8.9 8.8 8.7 8.6 0.0000119 .0000167 .0000226 .0000292 .0000367 8.6 0.0000446 logf Correction of logs, of S, n, f. 8.5 8.4 8.3 8.2 8.1 0.0000446 .0000525 .0000602 .0000672 .0000734 8.0 7.9 7.8 7.7 7.6 0.0000788 .0000835 .0000875 .0000909 .0000937 7.4 7.2 7.0 6.5 6.0 0.0000978 .0001006 .0001023 .0001044 .0001051 OO 0.0001054 The numbers in this table correspond to that state of the at- mosphere for which the refraction table (Table 11.) is computed ; that is, for the case in which the factors ^ and y of that table are each = 1. For any other case the tabular logarithm is to be varied in proportion to ^ and r- It is e\ident from this table that the eflTect of refraction will mostly be very small, for so long as the zenith distance of the moon is less than 70° we have log ^ > 9.53, and the tabular correction less than .000001. From the zenith distance 70° to 90° the correction increases rapidly, and should not be neglected. CORBECTION FOR THE HEIGHT OP THE OBSERVER ABOVE THE LEVEL OF THE SEA. 828. If 5' is the height of the observer above the level of the sea, it is only necessary to put p + s^ for /> in the general formulae of the eclipse ; and this will be accomplished by adding to log $, log ly, and log C the value of log 1 1 H — I, which is {M being the modulus of common logarithms) M L^-»(n-+H But s' is always so small in comparison with p that we mav 518 SOLAR ECLIPSES. neglect all but the first term of this formula ; and hence, by taking a mean value of p (for latitude 45°) and supposing ^ to be expressed in English feet^ we find Correction of log e, log 17, log C = 0.00000002079 8^ (565) For example, if the point of observation is 1000 feet above the level of the sea, we must increase the logarithms of f, 3;, and C by 0.0000208. If s' is expressed in metres^ the correction becomes 0.000000064 y. APPLICATION OF OBSERVED ECLIPSES TO THE DETERMINATION OF TER- RESTRIAL LONGITUDES AND THE CORRECTION OF THE ELEMENTS OP THE COMPUTATION. 329. To find the longitude of a place from the observation of an eclipse of the sxcn. — The observation gives simply the local times of the contacts of the disc of the sun and moon : m the case of partial eclipse, two exterior contacts only ; in the case of total or annular eclipse, also two interior contacts. Let w =r the west longitude of the place, t = the local mean time of an observed contact, fi = the corresponding local sidereal time. The conversion of t into /jl requires an approximate knowledge of the longitude, which we may always suppose the observer to possess, at least with suflicient precision for this purpose. Let Tq be the adopted epoch from which the values of z and y are computed (Art. 296), and let ^o» Vo = *^® values of x and y at the time T^y a/, 1/ = tlieir mean hourly changes for the time t -\- w; then, if we also put r = t + w-'T^ (566) the values of x and y at the time t +to (which is the time at the first meridian when the contact was observed) are The values of x' and y' to be employed in these expressions may be taken for the time t + to obtained by employing the LONGITUDE. 519 approximate value of oi, and will be sufficiently precise unless the longitude is very greatly in error. The quantities I and i change so slowly that their values taken for the approximate time t + co will not differ sensibly firom the true ones. For the same reason, the quantities a and d taken for this time will be sufficiently precise : so that, the latitude being given, the co-ordinates f , 7, ^ of the place of observation may be correctly found by the formulce (483). Since, then, at the instant of contact the equation (490) or (491) must be exactly satisfied, we have, putting L=l — if. 2/ sin Q = x^ — ^ -{- afr X cos Q = y^ — 1? + y'r } (567) in which r is the only unknown quantity. Let the auxiliaries niy My n, N be determined by the equations m%m M=zx^ — ? n sin JV:= j/ mco8Jf=y„ — 17 ncosiV=y' then, from the equations i sin C = wi 8i^ -3f + n sin JV . r L cos Q = m cos Jtf" -(- n cosiV. r by putting 4^=6 — iV, we obtain m sin (M — N) sm 4 = -z X cos 4* mooB(itf' — N) n n m sin {M — N — 4) n sin 4 } (568) (569) where the second form for r will be the more convenient except when sin 4^ is very small. As in the similar formulae (553), the angle '^ must be so taken that L cos 4^ shall be negative for first contacts and positive for last contacts, remembering that in the case of total eclipse X is a negative quantity. Having found r, the longitude becomes known by (566), which gives m = To — ^ + r (570) 520 SOLAB ECLIPSES. If the observed local time is sidereal, let fi^ be the Bidereal time at the first meridian, corresponding to 7^; then, r being reduced to sidereal seconds, we shall have and this process will be free from the theoretical inaccnracy arising from employing an approximate longitude in converting fx into L The unit of r in (569) is one mean hour ; but, if we write hLcos^ hmcos^M — J\r) n n - m sin (ilf — JV — 4») = A : i n sm^* we shall find r in mean or sidereal seconds, according as we take h = 3600, or A = 3609.856. 330. The rule given in the preceding article for determining the sign of cos '\^ (which is that usually given by writers on thw subject) is not without exception in theory, although in practice it will be applicable in all cases where the observations are suitable for finding the longitude with precision ; and, were an exceptional case to occur in practice, a knowledge of the approxi- mate longitude would remove all doubt as to the sign of the term — . But it is is easy to deduce the mathematical condition for this case. At the instant of contact, the quantity is equal to L^. At the next following instant, when r becomei» r + (h, it is less or greater than L^ according as the eclipse is beginning or ending. If then we regard i* as sensibly constant^ the differential coeflicieut of this quantity relatively to the time must be negative for first and positive for last contacts. The half of this coefficient is (X, - c + x^r) (a/- e') + (y, - ly + y'r) (y'- ,0 (where the derivatives of f and tj are denoted by c' and ij'), or, by (567), putting N + ^i^ for Q, L [sin {X + 4) (^ - c ') + cos (.V + 4) (^ - V)] LONGITUDE. 521 Computing f ' and yf by the formulae (551), or, in this case, by ^'= fi'p cos f ' cos (ji — a) rj'=: //( sin d and putting n' sin N'= of— f' n'cos N'= y'— V the above expression becomes in'cos(JV— i\r'+4) Hence, when L is positive, that is, for exterior contacts and interior contacts in annular eclipse, 4^ must be so taken that cos(iV— iV' + 4^) ^^^^^ he negative for first and positive for last contact. That is, for first contact 4^ ^^^ be taken between N' — N+ 90"" and N'—N+ 270''; and for last contact between If'—N+ 90° and N'—N— 90°. For total eclipse, invert these conditions. In Art. 322, we have N = iV', and hence the rule given for the case there considered is always correct. 331. To investigate the correction of the longitude found from an observed solar eclipse^ for errors in the elements of the computation. Let AX, Ay, Aiy = the corrections of x, y, and L, respectively, for errors of the Ephomoris, A?, Aiy = the corrections of S and rj for errors in p and f»', At = the resulting correction of r. The relation between these corrections, supposing them very small, will be obtained by differentiating the values of L sin Q and L cos Q of the preceding article, by which we obtain Aiy BinQ -\- L cosQ^Q = ax — a$ + a/ at Ai cos Q — iy sin § A § = Ay — Aiy -f y AT where ax and Ay, being taken to denote the corrections of X = x^,+ x'r andy = yo+ y'^, include the corrections of x' and y'. Substituting in these equations n sin N for x' and n cos N for y', and eliminating a§, we find aL = (ax — A$) sin Q + (Ay — Aiy) cos g + n cos (g — JV) . at and substituting for Q its value iV+ i^, . . sin (iV+ 4*) , .C08(JV+4') . ^i AT = — . (ax — Af ) ^ • — - — (Ay — Aiy) - — 5^ '—^ -I n COS 4^ n cos 4 n cos 4» 522 SOLAB ECLIPSES. or 1 1 AT = ( Ao? sin iV + Ay COS -AT) H — ( — aje; cos iV' -f Ay sin N) tan 4 1 1 -| — (a? sin JV4- Aiy cos iV) (— a? cosi\r4- A17 sin JV^ tan 4 /• fw + ^^ (571) which is at once the correction of r and of the longitude, since we have, by (570), aa; = at. 832. In this expression for at, the corrections ax, Ay, 4c. have particular values belonging to the given instant of observation or to the given place. In order to render it available for deter- mining the corrections of the original elements of computation, we must endeavor to reduce it to a function of quantities which are constant during the whole eclipse and independent of the place of observation. For this purpose, let us first consider those parts of at which involve ax and Ay. For any time 7^, at the first meridian, we have y =y.+ n COB NiT.^T,) whence X sin JV -|- y cos JV = x,, sin iV^ + y„ cos JV4- n ( T^ — T^ — X cos iV + y sin JV = — Xq cos iV + y^ sin iV The last of these expressions, being independent of the time, is constant. If we denote it by x ; that is, put X = — Xq cos iV -f y^ sin JV= — x cos N + y sin N (572) we shall find from the two expressions xx + yy = x>c + [x^sin JVr+ y.cos iVr+ n(7;— T.)]' (573) This equation shows that the quantity |/xx + yy, which is the distance of the axis of the shadow from the centre of the earth, can never be less than the constant x, and it attains this minimum value when the second term vanishes, that is, when x.sm N + y^cos N + n(^T,^ T.) = and hence when T, = T, — 1 (Xosin N+ y„cos iV^ (574) LONQITUDE. 523 which formula, therefore, gives the time 7\ of nearest approach of the axis of the shadow to the centre of the earth, while (572) gives the value of the distance of the axis from the centre of the earth at this time. By the introduction of the auxiliary quanti- ties 7\ and x, we can express the corrections involving ax and Ay in their simplest form ; for we have now, for the time of obser- vation < + a>, a: sin JV + y cos J\r:= x^ sin iV -f y^cos iV -|- » (^ -f a> — T^ = n (t + a» — T,) and if An, aTj and ax are the corrections of w, 2\, and x on account of errors in the elements, we have AX sin JV+ Ay cos N= — n aT^ + (^ + a» — 2^) An 1 — AX cos iVT 4- Ay sin JV^ = AX / ^^ ' ^^ These expressions reduce those parts of at which involve ax and Ay to functions of a Tj, am, and ax, which may be regarded as constant quantities for the same eclipse. We proceed to consider those parts of at which involve a^ and ^yj. These corrections we shall regard as depending only upon the correction of the eccentricity of the terrestrial meridian ; for the latitude itself may always be supposed to be correct, since it is easily obtained with all the precision required for the calculation of an eclipse ; the values of a and d depend chiefly on the sun's place, which we assume to be correctly given in the Ephemeris ; and [i is derived directly from observation. Now, we have (Art 82), e being the eccentricity of the meridian, cos cp . , (1 — ee) sin w l/(l — ee sm' ^ ) l/ (1 — ee sm' ip) whence, by diflferentiation, A/ocosy^ _ , pp sin* y^ = p cos • Aee '^ 2(1 — ee)« A./psiny^ . , pp sin' y^ p sin y^ = P Sm • r — Aee ^ 2(1 — eey 1 — ee or, putting P sin ^' /5 = 1 — ee 524 SOLAR ECLIPSES. A./9 COS / A . /> sin ip* From the values = } ^^p cos 9/ = ipfipAnf/'^fi (576) S = p cos ^' sin (;ei — a) 19 = /> sin f ' cos 4 — p cos 9/ sin (2 cos (;* — a) we deduce and hence A^sin N+ A7C08iV=}^^( f sin iV + 9 C08 iV') Am — )3 eos cf cos i\r Am — A^ cos iV -f- A7 sin iV = J j3/3 ( — ^ cob H+if sin iV) Am — fi cos (i sin i\r Aee The values of ^ and 37 may be put under the forms f = x^— (x^— f) = rTo— m sin ilf ^ = yo — (yo— 'y) = yo— wi cos jf by which the second members of the preceding expressions are changed respectively into i P^ {. ^0 ^^^ ^ + y©^®* ^ — "* ^®* (^ — ^)^ ^^^ — P cos d COS A*" Am and i pp [— ZqCOS N -\- y^Bin N -^ m sin (i/ — iV)] Am — ^ cos d sin JV Am or, by (574) and (572), into i PP [n (TJ, — 7\) — TO cos (M— iV)] Am — P cob d cos iV Am and \ PP\_ X -\- mBia (M — iV)] Am — /? cos <2 sin iV Am or, again, by (569) and (570), into ^ Pp\n(t -\-ti— T^) — L cos 4*] Am — P cob d cos iV Am and h PP\. X -\- L Bin -^l Am — p cos d sin iV Am Hence, that part of at which depends upon ^ee is equal to PP r f* I m\ ^ , r nx 5 COS d COS (iV -f 4) ^ [n (« 4- w — 7\) — X tan "4/ — /i sec 4] Am — t:^ ^ !— ^ Am 2n n cos 4* When these substitutions are made in (571), we have Ar = Aw = Aa7\ -f A tan 4 . ^ — A (< 4- 0) — Tj) — + A sec 4/ . — n n « + - Fi-^/? [» (« + " - r.) - » tan 4 - Z sec 4] - ^£2L^i^iH'±±n A« (577) W L C03 4 J LONGITUDE. 625 where we have multiplied by h to reduce to seconds. The unit is either one second of mean or one second of sidereal time, according as r is in mean or sidereal time. If the former, we take h = 3600; if the latter, h = 8610. 333. The transformations of the preceding article have led us to an expression in which the corrections a 7\, ax, ati, and ^ee are all constants for the earth generally, and which, therefore, have the same values in all the equations of condition formed from the observations in various places. But a still further transform- ation is necessary if we wish the equation to express the rela- tion between the longitude and the corrections of the Ephemeris, so that we may finally be enabled not only to correct the longi- tudes, but also the Ephemeris. Since a7\, ax, aw are constant for the whole eclipse, we can determine them for any assumed time, as the time 7\ itself. For this time we have x sin JV + y cos N=0 — a:cosiV4- y8iniV = x AX sin iV + Ay cos N= — n a2\ ^ (^^^) — AX cos iV + Ay sin JV = ax The general values of x and y (482) may be thus expressed : JT Y ^ = ^T;r- y sm TT "^ smn where X= cos ^ sin (o — a) F=: sin dcosd — cos ^ sin d cos (o — a) From these we deduce aX att aF A?r ^^ = ^t;;^ - ^ T^;rz ^y = ^i:rz-y smir tauTT ^ sinff ^ tanw whence Az sm iV 4- Ay COS iv = ! (« Bin JV + y cos JV) sin ff tan ir Air -AX cosiV+ Ay sin JV= ~^^cosiV+AFsiniV ^^ cos JV- y sin N) sinff ^^ ^ ^tanff and for the time T^ these become, according to (578), ^rn A-tTsin JV+ aFcos JV * sm n — A J!" COS N+ A Fsin N a^ AX = . — ! X sm T tansr 524 SOLAR ECLIPSES. A . p COS <p' A . p sin ^' Aee From the values = } pfip cos f ' = }i9;9/9 8insp'--i9 (576) ^ = p cos f»' sin (;ei — a) iy = /t) sin f ' cos d — p cos 9/ sin d cos (;* — a) we deduce ^=}^^e -^=}^/?iy-/9cos<f and hence Af 8iiiiV+ A^C08iV=}^^( f 8iiiiV+ J7C08iV) A^e — )3 cos </ cos A^ A«r — Af C08iV-f- A^sin iV= J ^^ (—^ cos iV+ 17 sin iST) Aee — /? cos <2 sin .V Am The values of f and 7 may be put under the forms $ = ar^,— (^o~ = ^0— ^^ sin 3f ^ = yo — (yo — ^) = yo — ^ cos jtf" by which the second members of the preceding expressions are changed respectively into J P? [ ^Q^^^ N" 4- yo<i09 N — m 008 (J/ — iV)] Ar? — (i cos */ cos .V A^f and J /?/?[— Zq cos iV -f- yosin iV -f- m sin (i/ — iV)] Aw — ,? cos d sin ->' A« or, by (574) and (572), into } PP \n {T^ — Tj) — m cos (M — N)^ Arr — /? cos d cos X A« and }/?/?[ X -f m sin (AT — iV)] A« — p coa d sin A" Af« or, again, by (569) and (570), into i Pl^ [n {t + u— T^) — L cos 4] Am — /? cos rf cos A' Afif and i ft? [ X -}- /y sin 4] ^^'^ — /? co8 ^^ sin A'' Aw Ilence, that part of at wliich depends upon ^ce is equal to Pt^ r /* , m\ * , r , -i a /? COS </ COS (A' -f «^> ^ ^ [n (t + u — TJ) — X tan 4/ — /^ sec 4/] Aw — ^^ ^^ l_I- ^(f 2n n COS ^ When these substitutions are made in (571), we have Ar = Aw = Aa7\ + A tan 4/ . ^ — A (< + o — r,) — -f A sec 4 . — fi n n , ^Fioor /- I /n\ 1 • r .t /3 COS rf COS (A' - -^ )"| . -f - J /?;?[»(< + w — 7\) — X tan 4 — Jy sec 4] — ^- i ^ I Aw ^oi • ) LOXGIirDK. '^-"^ where we have multiplied bvA to reduce x? fe: r:.^!-. Tlo \:v.:: is either one second of mean or onr >ec":'i:'i oi ?::oro;r: v.'.r.o. according as r is in mean or sidereal r.ii.e. K tLe iori:»«.r, wo take h = 3600; it* the laner, k = ot.lO. 333. The transformations of the preoed:nir article luwo Ud v.s to an expression in which the correot:oL> aT. ax. a;i, aiul ai aiv all constants for the eanh ^enerallv. and which. thcr*.'tori\ luwo the same values in all the equations oi condition lorniod tVviu the ohservations in various places. But a still further transtoini- atiou is necessarj- if we wish the e4uation to express tlio rela- tion between the longitude and the corrections of the Kphonieris, 80 that we may finally be enabled not only to correct tlio louiri- tudes, but also the Ephemeris. Since aT,, ax, a?i are constant for the whole eclipse, wo can determine them for any assumed time, as the time 7; itself. Kor this time we have X sin X + y cos iT = XQOSX+ ysin y = x Aj sin T 4- Ay cos JV = — n A r -a:co8.V+ ysin y = x \ .. , — AJcosJV-p Ay sin jV= AX ) The general values of x and y (482) may be thus exprcssevl: sinw ^""Bin:^ where ^= cos ^ sin (a - a) 7= sin a cos d - cos 5 sin d cos ^« '-^ Prom these we deduce A-*! A;r a ir v- AX = r - ^' "•• whence ""^^ ^^''' ^y = ^->'u.ur: sin TT Bin T — Azco8.V + AyBin^':=,:3A£co8iV4-Ar8miV .. ■.. V) . ^j + (x COS N y »*'" '^ !#» ' and for the time T Watx^ v ^itJiese become, according to (57H), -nAT,=^£^n^+ aYcos .V sm TT Bin 7 tiiii jr 526 SOLAR ECLIPSES. Again, by differentiating the values of X and F, we have A JT := COB ^ cos (o — o) A(a — ol) — Bin ^ sin (a — a) A* A Y = [cos ^ cos d '\- WDih Bin d cos (a — a)] a^ — [sin ^ sin (Z + cos ^ cos d cos (a — a)] a4 + cos d sin (2 sin (a — a) A(a — a) But for the time of nearest approach we may take a = a ani) put cos (5 — d) = 1, whence A-r= cosd.A(o ^-a) aF= a(^ — d) so that _ sin i? cos d . A(a — a) + cos iV . A(d — d) — nA i ,== : sin iz : (679) — oos-ZVcos ^. A(o — a) -|- sin JV. A(^ — d) Ax Ax == --— ^-^---— ——-----— —--y-— ---——— ————— M -►^— sin TT tan -k To find An, which depends upon the corrections of 7f and y, we observe that x' and y', regarded as derivatives of x and y, are of the form ^ dX \ , dY \ dT sin^r dT sinx But -j=^ and -^^ depend upon the changes of the moon's right ascension and declination, which for the brief duration of an eclipse are correctly given in the Ephemeris. The errors of t! and y\ therefore, depend upon those of;:: so that if we write sm n sm TT and regard a and h as correct, we find Ax'=~a/ ^}/ '= — y tauTT tauTT From the equations n sin iV= a:', n cos iV= y', we have A9r An sin iV4- n AiVcos JV= Aa/= — nsin iV An cos JV— n AiVsin J\r=rAy'= — ncosJV tan?r ATT taUTT LONGITUDE. 527 whence, by eliminating ^N^* :^ = ^^ (580) » tan n Since A(a — a), ^{d -— rf), A;r will in practice be expressed in seconds of arc, we should substitute for them a (a — a) sin 1", a(* — d) sin 1", A;r sin 1'' in the above expressions ; but if we at the same time put it sin 1" for sin tt and tan «■, the factor sin 1" will disappear. To develop aZt, we may neglect the error of the small term i ^ and assume aX = a?. We have from (486) and (488), by neglecting the small term k sin tt^ and putting ^ = 1, z = -: y the following approximate expression for I : , sin JJ , 1 = -— — ±k rsmTT which gives Ai == aZ = -^ =b A* -« — . — (681) Substituting the values of aT^i, ax, ah, and a{ given by (579), (580), and (581), m (577), and putting nit the formula becomes, finally, Au=. — y[ sin JVoo8dLA(a — a) -|- eo8 JV.A(<5 — d)} + V [— 008 iVeos d.A(a — a) -|- sin y,^{6 — d)] tan 4 + v| —;- =tZ IT A* I 800 4 ] + v|ii (< 4- « — 7\) — » taii4 — -j-boo^Iait + v[}i5^[i»(< + «-r,)-»tan + -2i8004]-^^5li^^!^ (582) where the negative sign of «:aA: is to be used for interior contacts. It is easily seen that n^k represents very nearly the correction * The angle iVis independent of erron in ir, iinoe tan iVcs >: so that we might haTe taken AiV=0. 528 SOLAR ECLIPSES. of the moon's apparent semidiameter, and —j- that of the son'g semidiameter ; and that nt^e is the correction of the assumed reduction of the parallax for the latitude 90°. 334. Discussion of the equations of coTuUtion for the ccrrectim of the longitude and of the elements of the computation. — The longitude w found by the equation (570), (Art 329), requires the correction ^Q) of (582). If, for brevity, we put y = Bin N cos d A(a — «) + COS JVa(^ — d) 1 (^ag\ ^ = — cos iV COS d A(o — a) + sin N^(d — d) i and ctf'= the true longitude, we have the equation of condition ai' = 01 -f Aai = Ai — v/* + y tan 4 . ^ -|- &c. (584) If the eclipse has been observed at several places, we can form as many such equations as there are contacts observed. If the observations are complete at all the places, we can, for the most part, eliminate from these equations the unknown corrections of the elements, and determine the relative longitudes of the sevenl places ; and if the absolute longitude of one of the places is known, that of each place will also be determined. I shall at first consider only the terms involving y and i?. The quantity v;' is a constant for all the places of observation, and combines with w, so that it cannot be determined unless the longitude of at least one of the places is known. If then we put Q = at' -\- vy a = V tan 4/ the equations of condition will assume the form fl _ rt,9 _ a» = Suppose, for the sake of completeness, that the four contacts of a total or annular eclipse have been observed at any one place, and that the values of the longitude found from the several con- tacts by Art. 329 are w^, w^, cw,, o)^. We then have the four equa- tions [1] fl — a^ 1^ — a»j = [2] fl — a, * — o», = [8] C — a, 1^ — «, = [4] C — a^i^ — «^ = LONGITUDB. 529 where the ntimerals may be assumed to express the order in which the contacts are observed ; [1] and [4] being exterior, and [2] and [3] interior. In a partial eclipse we should have but the 1st and 4th of these equations. Since exterior contacts cannot (in most cases) be observed with as much precision as interior ones, let us assign different weights to the observations, and denote them by je>i, p^y p^j p^, respectively. Combining the four equations according to the method of least squares, we form the two normal equations [p ] fl — . [pa ] »» — Ipw ] = Ipa] Q — Ipaa] & — [pao*] = where the rectangular brackets are used as symbols of smnma- tion. From these, by eliminating £, and putting we find P,> -f Q = (585) from which the value of t> would be determined with the weight P. But the computation of Q under this form is inconvenient. By developing the quantities P and Qj observing that [j>aa] = Pfii + P^ + P^ + PfiiJ ^^'j we shall find P— PiPi {^i — «»)* + Pi Ps (^i — ^'a)' -h Pi Pi («i — «4)* Pi -\- Pt -{- Ps -\- Pi I Pi P% (g» — «»)* -f Pt Pi {^i — g*)' + Ps Pi (Oj — ^4)* Pi+Pt-^-Pz +Pa Q ^ P\ Pt (<'i — «») ('■^i — "2) + Pi Ps (fli — gg) ("1— s) -f Pi Pi (gj — Oj) K — s) Pi -i-Pi-^Ps+Pi , PtPt (g» — 09) (S — S) + Pt Pi i^i — «<) (^2 — "4) + PsPi (^5 -- ^4) (S - ^4) Pi-^-Pt+Ps-^Pi These forms show that if we subtract each of the equations [1], [2], [3] from each of those that follow it in the group, whereby we obtain the six equations (aj — a,) 1? -f oij — 01, = («i — a,) * + a*j — «/, = (aj — a J iJ + CI*, — ci*^ = («. — «») t> + cii^ — 013 = («« — aj ^ + a*. — 01^ = (d. — aj !?+.«;, — 01^ = Vol. I.— 34 530 SOLAR ECLIPSES. and combine these six equations according to the method of least squares, taking their weights to be respectively VxP. P.P. ,4^ Vi+p» + ;>a +i>4 Pi+p»+ Pt+P4, we shall arrive at the same final equation (585) as by the direct process, with the advantage of avoiding the multiplication of the large numbers w^^ a;,, &c. Suppose that at another place but three contacts have been observed, the true longitude being w''^ and the computed longi- tudes oij, w^j fi>y, and that, having put i2'=ctf"+ vy^ we have formed the three equations [5] Q! — a, i> — fl'ft = with tlic weight p^ [6] iy— .dji* — tt*j = " " p, [7] fi'— ay*— flly = " " pj The subtraction of each of the first two from those which follow gives the three equations («5 — «6) * + "'s — "'e = (a* — a,) * + fi*, — "'y = ^ («e — ^) ^ + fi'e — "'j = ^ of which the weights will be respectively, according to the above forms, PsP^ Pi Pi PtPi and the combination of these three equations, according to weights, will give a normal equation of the form which gives a value of # with the weight P'. Now, suppose that this method applied to all the observations at all the places has given us the series of equations in i?, P»!> + Q =0 P tJ + Q' = P".^+ Q"=0,&c.; then, since P, P', P", &c. are the weights of these several deter- minations, the final normal equation for determining tf, derived from all the observations, is XONGITUDB. 631 that is, it is simply the sum of all the individual equations in & formed for the places severally. The same reasoning is applicable to any of the terms which follow the term in t? in (584) ; so that if we suppose all the terms to be retained, this process gives an equation in t? for each place, in which besides the term P9 there will be terms in aA:, ^H, &c., and from all the equations, by addition, a final normal equation (still called the equation in d) as before. In the same manner, final normal equations in aA:, a1?, &c. will be formed. Thus we shall obtain five normal equations involving the five unknown quantities t?, aA:, aJJ, A;r, Ae6, which are then determined by solving the equations in the usual manner. But, unless the eclipse has been observed at places widely distant in longitude, it will not be possible to determine satisfactorily the value of ^jtj much less that of Me. It will be advisable to retain these terms in our equations, however, in order to show what effect an error in tt or ee may produce upon the resulting longitudes. When I?, &c. have been found, we find J2, £', &c. from the equations [1], [2] . . . . [5], [6] The final value of S will be the mean of its values [1 — 4] taken with regard to the weights ; and so of £', &c. Hence we shall know the several differences of longitude w — a,"= Q-^Qf, a,' — a,'"= fi — fl", &c. K one of the longitudes, as for instance (o'y is previously known, we have and hence all the longitudes become known. Finally, from the values of y and t> the corrections of the Ephemeris in right ascension and declination are obtained by the formulfle cos ^ A(a — a) = sin iV. y — cos N , ^ 1 r5S6> A(5 — (f) = C08iV.;'+ sin iV.d I ^ 335. When only two places of observation are considered, one of which is known, it will be sufficiently accurate to deduce y and t> from the observations at the known place (disregarding the other corrections), and to employ their values in finding the lou^tude of the other place. 532 SOI4AR ECLIPSES. 336. TVTien good meridian observations of the moon are avail- able, taken near the time of the eclipse, tlie quantities A(a — a\ a(5 — d) [for which we may take A(a — a'), a(5 — 5')], may be found from them. The terms in y and «> may then be directly computed by (583) and applied to the computed longitude ; after which the discussion of the equations of conditiou may with advantage be extended to the remaining terms. 337. Before proceeding to give an example of the computation by the preceding method, it will be well to recapitulate the necessary formulse, and to give the equations of condition a practical form. I. The general elements of the eclipse, a, d, /, log i, ar, y, x', y, are supposed to have been computed and tabulated as in Art. 297. n. The latitude of the place being ^, the logarithms of p cos f ' and p sin f' are found by the aid of our Table HI., or by the fomiulie (87). The mean local time t of an observed contact being given, find the corresponding local sidereal time p ; also the time ^ 4- « at the first meridian, employing the approximate value of the longitude (o, [If the observed time is the sidereal time /i, the time /£ -f « at the fii^st meridian, converted into moan time, will give the approximate vahic of t + to,'] For the time t -{■ (o take a, d, Z, and log i from the eolipe tables, and compute the co-ordinates of the place and the radium of the shadow by the formulte A i^\n B = p sin 9"' A cos B ^^ p cos f'cos (fjL — a) ^ =P cos f ' sin (fi — a) jj = A sin {B — d) Z =Acos{B — d) AVhon log ^ is small, add to log ^, log 3j, and log ^ the correc- tion for refraction, from the table on p. 517. III. For the assumed epoch 7i at the first meridian (being the epoch from which the mean hourly changes jr' and y' are reck- oned), take the values of x and y from the eclipse tables, denoting them by 0*0 and y^. Also the mean hourly changed x* LONGITUDK. 533 and y' for the time t + o}. Compute the auxiliaries m, M, &c. by the formulse* mmi M =x^ — f n sin i\r = a/ m cos M= y^ — 17 n cos N = y' m sin fJlf — N) sm <4' = ^^ where 1^ is G'^ general) to be so taken that L cos ij/ shall be negative for a first and positive for a last contact (but in certain exceptional cases of rare occurrence sec Art. 330). Then hL cos 4 Am cos (M — N) n n or, when sin ^ is not very small, hm sin(Jtf" — N — 4) ^ sin 4 If the local mean time t was observed, take h = 3600 in these formulae, and then the (uncorrected) longitude is found by the equation oi = 7; - f + r If the local sidereal time /i was observed, take h = 3609.856, in the preceding formulae ; then, /i^ being the sidereal time at the first meridian corresponding to TJ, we have at = /*o — /* + ^ The longitudes thus found will be the true ones only when all the elements of the computation are correct IV. To form the equations of condition for the correction of these longitudes, when the eclipse has been observed at a suffi- cient number of places, compute the time 7J of nearest approach, and the minimum distance x, by the formulae 2; = 7; - 1 (X, sin N+y, cos N) X = — x^ COS i\r + y© ^^^ -^ * The yalues of JV and log n being nearly constant, it will be expedient, where many obserrations are to be reduced, to compute them for the several integral hours at the first meridian, and to deduce their yalues for any given time by simple interpolation. 534 SOLAR ECLIPSES. Take k for the time 7\, and compute the logarithm of the same value of h heing used here as before. For each observation at each place compute the coefficients V tan 4') ^ sec ^^^ &ud E = vnCt +at — T.) — XV tan 4. r- »* sec 4 where the unit of / + a; — T^i is one mean hour, r» 1 OiO r /J I m\ * • r n Vi3 COS rf 001 (A' -1- 4) F= ipp [vn(< -f « — 7\) — XV tan <4' — 2^ v sec <4'] i 2I 000 4 in which J = 959".788 log H = 2.98218 ^ ^ ^osin^' J _ ^ 9.99709 1 — ee Then, a>' denoting the true longitude, the equation of condition is L>'= (J — v.y + v tan 4'*v=zzy8ec4'*^Ait-|-v8ec<4'* h -i?Ajr -f /*. t A« where the negative sign of the term v6ec4'«^AA: is to be useil for interior contacts. The discussion of the equations thus formed may then be carried out by Art. 334; taking as the unknown quantities Y, t?, TT^k, —J- J A'T, and TZMe, Example. — ^Find the longitude of Washington from the fol- lowing observations of the solar eclipse of July 28, 1851 : At Washington (partial eclipse) : Beginning of eclipse, July 27, 19* 21- 31v2 M.T. End " « " 20 50 38.0 " At Konigsberg (total eclipse) : • Beginning of eclipse, July 28, 3 38 10 .8 « Beginning of total obsc, '' ** 4 38 57 .6 " End of total obscuration, " " 4 41 54 .2 " End of eclipse, " " 5 38 32 .9 " LONGITUDE. 535 For these places we have given — Lai. ^ Washington, + 88^ 53' 39".25 Konigsberg, +54 42 50 .4 Long, o + 5» 8- 11'.2 — 1 22 0.4 The longitudes are reckoned from Greenwich. That of Konigsberg will be assumed as correct, while that of Washington will be regarded as an approximate value which it is proposed to correct by these observations. I. The mean Greenwich time of conjunction of the sun and moon in right ascension being, July 28, 2* 21* 2'.6, the general eclipse tables will be constructed for the Greenwich hours 0*, 1*, 2*, 3*, 4*, and 5* of July 28. For these times we find the follow- ing quantities from the Nautical Almanac : For the Moon.* Qreenwich mekn time. a d TT July 28, 0» 125*^40' 6".75 + 20^ 3'30".00 eC 27".30 1 126 19 9 .41 19 58 9 .86 28 .41 2 126 58 10 .80 19 52 39 .99 29 .49 3 127 37 10 .82 19 47 1 .92 30 .54 4 128 16 9 .37 19 41 15 .21 31 .56 5 128 55 6 .36 19 35 19 .89 32 .56 For the Sun. Greenwioh mean time. o' i' logr- July 28, 0» 127° 6' 5".25 4- 19° 5' 24".70 0.006578 1 8 32 .63 4 50 .28 76 2 10 59 .99 4 15 .74 74 3 13 27 .34 3 41 .21 72 4 15 54 .67 3 6 .64 70 5 18 21 .99 2 32 .05 67 * The moon's a and 6 in the Naut. Aim. are directly computed only for every noon and midnight and interpolated for each hour. I have not used these interpolated Talues, but have interpolated anew to fifth differences. The moon's parallax has been diminished by (y\Z according to Mr. Adams's Table in the Appendix to the liaut. Aim, for 1856. 536 SOLAR ECLIPSES. "Witli these values we form the following tables, as in Art 297 : 0* a d KxterioT Oontects. Interior CoBtarta. I log* I loc< 127® 6' 17".22 19® 6' 16".66 0.684046 7.663244 — 0.011771 1 7.661131 1 8 89 .61 4 42 .76 4028 46 11795 82 1 2 11 1 .78 4 8 .96 8978 47 11844 34 8 13 24 .03 8 86 .14 8899 49 11917 3»; 4 16 46 .27 8 1 .80 8801 61 12016 88 6 18 8 .60 2 27 .46 8679 68 12137 40 0* 1 2 8 4 6 X A» + 67 1 — 66 —119 A, —68 —64 -64 y A. A. 1 1.888900 — 0.769366 — 0.199776 + 0.869816 + 0.939860 + 1.608766 + 0.669684 .669691 .669590 .669636 .669416 + 0.968689 .885669 .802185 .718449 .684870 .649950 — 0.083020 .083884 .083736 .084079 .084420 -864 —352 —843 —341 -rl2 J- 9 •» 1 I Hence the mean changes r' and y\ for the epoch 7i = 2* (ac- cording to the method of Art. 296), and the corresponding values of N and log n, are as follows : x' y' JV logii 0* + 0.569563 0.083202 98° 18' 39".7 9.700126 1 591 3384 19 42 .7 16S 600 3562 20 45 .3 194 3 590 3736 21 47 .5 205 4 563 3908 22 50 .0 203 "> 514 4078 23 52 .7 186 II. Tlie full computation for Konigsborg, where both exterior and interior contacts were obsorv^ed, will ser\'e to illustrate the use of the preceding formula^ in every practical case. For ^ = 54° 42' 50".4 we find log p Bin sp' = 9.909898 log p cos f ' = 9.762639 The sidereal time at Greenwich mean noon, July 28, was 8* 22"* 13\27, with which /i is found as given below. The com- putation of c, jy, and L will be as follows : «« LONGITUDB. 687 lit Sxt. Cont. lit Int. Cont. 2d Int. Cont. 2a Ext. Cont. t 8* 88- IC.S 4* 88« 57'.6 4* 41- 54'.2 5* 38"« 32'.9 t + u 2 16 10.4 8 16 57.2 8 19 53.8 4 16 32.5 ^ 12 46.44 18 1 43.22 18 4 40.81 14 1 28.31 fi (in arc) 180® 11' 86".6 195® 25' 48". 8 I960 10' 4".7 210«22' 4".7 For < + «, a 127 11 40 .1 127 14 4.2 127 14 11 .2 127 16 25 .6 " d 19 8 59 .8 19 8 25 .6 19 3 28 .9 19 2 52 .0 fi^a 52 59 56 .5 68 11 44 .1 68 55 53 .5 83 5 39 .1 log sin (fi — a) 9.902848 9.967762 9.969952 9.996838 log C08 (/I — a) 9.779473 9.569889 9.555679 9.080040 log^ 9.664982 9.730401 9.732591 9.769477 ^ + 0.462862 + 0.587528 + 0.540244 -f 0.574748 log Asm B 0.909898 9.009898 9.909898 9.909898 log ^ 008 ^ 9.542112 9.832528 9.318318 8.842679 5 66<» 47' 82".2 75« 10' 40".4 75<>38' 5".9 85* 6'14".3 B — d 47 48 82 .4 56 7 14 .8 56 34 42 .0 66 8 22 .3 log A 9.946544 9.924595 9.923693 9.911486 log sin (5 — d) 9.869192 9.919191 9.921499 9.960919 log COS (^ — d) 9.827809 9.746201 9.740991 9.608855 lOgff 9.815786 9.843786 9.845192 9.872405 n + 0.654289 + 0.697888 + 0.700152 + 0.745427 logC 9.774858 9.670796 9.664684 9.519841 ' < 4- «, log i 7.668248 7.661137 7.661187 7.663262 " / -f 0.538956 0.011940 0.011944 -f- 0.533772 K + 0.002789 H- 0.002148 -f 0.002117 4- 0.001524 L + 0.531217 0.014088 0.014061 -f 0.532248 in. The epoch of the table of x' and y' being 7^= 2*, we have for this time x„ = — 0.199775 y, = + 0.802185 with which we proceed to find the values of w. m sin if — Xq — ^ 0.662137 0.737303 0.740019 — 0.774523 « cos -¥ — yo — 7 4- 0.147946 -f 0.104297 + 0.102033 4- 0.056758 log m sin M n9. 820948 119.867646 n9.869242 n9.889035 log m cos M 9.170107 9.018272 9.008741 8.754027 M 282« 35' 42". 8 278<> 3' 5".4 277«51' 1".5 274« 11'28".3 log m 9.831527 9.871949 9.873381 9.890198 Por < + «, N 98<>21' 2".l 98«22' 6".l 98«22' 8".2 98<>23' 7".3 " " log n 9.760198 9.760206 9.760206 9.760200 M—N 184<»14'40".7 179<»41' 0".3 179* 28' 53".3 176« 48' 21".0 log sin (M — .V) n8. 869321 7.742368 7.956643 1 8.864135 588 SOLAB ECLIPSES. logL log Bin 4 log8in(if— A^— 4) A = 3600, log A logr T 9.725272 118.976576 186« 26' 27".7 368 49 13 .0 f»8.318626 3.566303 2.966682 4- 0* 16- 24^.0 n8.148849 R9.466468 343« 1' 8".6 196 39 61 .7 119.467626 3.660109 -f 1* 10-« 12'.0 118.148016 i fi9.681958 j 208O 44' 14^.0 9.726114 9.028219 6<> 7'Sr.2 330 44 39 .8 1G9 40 47 .8 n9.689051 8.676621 9.263208 3.911290 + 1*19* S-.i; 4-2*15-o2'5 — 1 38 10.8 —2 38 57.6 —2 41 64.2 —3 38 32.9 — 1 22 46.8 —1 22 45.6! — 1 22 46. ij —1 22 40.4 IV. JSquations of condition. — To find T^ and x, we have for log x^ = n9.3006 logyo= 9.9043 whence J\^=98^20'.7 log n = 9.7602 x^ siniV n l/o COS N n = + 0.3434 = + 0.2023 T; = 2*.5457 7r= 3630" — x. cos JNr= — 0.0290 log J =2.9822 + y^BinN= + 0.7938 X = + 0.7648 log X = 9.8835 logf logr log — = 0.0066 = 3.5599 = 9.4157 log fi = log ^-^^ = 9.9128 1 — ee log y = log -^ = 0.2362 ;i With these constants prepared, we readily fomi the cooflBcients of the equations of condition as follows : l8t Ext. Cont. 1 1st Int. Cont. 1 2d Int. Cont. i adKzt.Cunt. log tan 4 8.9775 1 W9.4848 9.7390 9.aW7 log sec 4 n0.0019 ' 0.0194 «0.0671 O.0rr25 V tan 4 -f 0.163 ' —0.526 + 0.944 -f 0.185 V sec 4 1.730 i -f- 1.801 — 1.964 -r 1.738 t-ru-r, 0*2762 -f 0*.7355 -f 0».7860 -: l*.73n<) log (ti-u-T,) n9.4412 9.8666 9.8954 0.2380 vn{t^u- 7\) 0.2739 -f 0.7295 -i- 0.7795 -^ 1.7155 — XV tan 4 - 0.1251 -f 0.4023 — 8.7228 : — 0.1414 II V sec a r'jT -r 0.4506 — 0.4691 4-0.5117 — 0.4512 J?, -^ 0.0516 -i 0.6627 -u 0.5689 ' - 1.1229 LONGITUDS* 589 — XV tan 4 — Lv 8604 log log}/?/? log let pari of F log cos (iV + 4) log ( — vp cos d sec 4) log 2d part of F Ist part of F 2d •I •• Jf Ut Sxt Oont. Itt Int. Cont. SdlntOoot 2d Ext. Oont 0.2789 — 0.1261 + 0.9192 -f 0.7295 + 0.4028 + 0.0264 4- 0.7796 — 0.7228 — 0.0276 -f 1.7166 — 0.1414 — 0.9222 4- 0.6202 9.7162 9.6246 + 1.1672 0.0684 -1- 0.0296 8.4718 -f 0.6619 9.8142 9.2408 288* 46'.2 9.8766 0.1264 9.6880 81*» 2r.8 9.1766 ftO.1489 7.9969 807« 4'.9 9.7808 0.1816 9.3888 104» 28'.8 n9.3978 ftO.1270 9.6030 -f 0.1741 -f 0.8184 n9.8206 + 0.8878 — 0.2092 9.9619 + 0.0099 + 0.9160 9.6248 + 0.2182 -f 0.8849 -f 0.4926 H- 0.1781 + 0.9269 + 0.6581 Putting a»' + v;* = J2, we have, therefore, for the four Kouigs- berg observations, the equations (A) p 1 o at 2 1 0=— 1* 22«46«.8 + 0.168 1^ — 1.730 TTA*— 1.780 — +0.062 A7r + 0.498ffAw 0=— 1 22 46.6—0.626 —1.801 +1.801 +0.668 +0.178 = — 1 22 46.1+0.944 +1.964 —1.964 +0.669 +0.926 0=— 1 22 40.4 + 0.186 +1.788 +1.788 +1.128 +0.668 where we have annexed a column for the weight p, giving interior contacts double weight. A similar computation for the two observations at Washington gives the following equations, in which S'=^ w" + vj^ cw" de- noting the true longitude of Washington : (B) 1 1 0* = 6» 7-« 29».9 + 1.660 ^ — 2.8927rA* — 2.892 — — 2.681 Att + 0.722 n-Af* 0^=6 7 21.9 — 2.406 +2.969 +2.959 +0.609 —1.828 More observations would be necessary in order to determuic all the corrections ; but I shall retain all the terms in order to illustrate the general method. Subtracting each of the Konigs- berg equations from each of those which follow it, we obtain the six equations, 540 SOLAR ECLIPSES. (A') P i i i i i i = 4- 1'.2 — 0.689 & — 0.071 tA* + 3.531 — + O.Cll Ajt — 0.315tAm = + .7 -f 0.781 + 8.694 — 0.284 + 0.617 -f 0.483 = -f 6 .4 + 0.022 -f 8.468 + 8.463 + 1.071 -f 0.060 = — .6 4- 1.470 + 3.765 — 8.765 — 0.094 + 0.748 = 4-5.2 + 0.711 4-8.534 —0.068 4-0.460 4-0.375 = -f 5 .7 — 0.759 — 0.281 4- 8.697 4- 0.554 — 0.373 where the weight in each case is the quotient obtained hy dividing the product of the two weights of the equations whose difterence is taken, by the sum of the weights of the four original equations (Art 334). The same method, applied in the case of the two Washington equations, gives the shigle equation (B') ^ff = — S'.O — 4.066 i» 4- 5.851 ttA* 4- 5.851 — 4- 8.190 At — 2.055 tAi« From the equations (A') and (B') are formed the following final equations, having regard to their weights, in the usual manner : = 4- 15.495 4- 10.426 1» — 5.300 tA* — 16.377 — — 6.609 At 4- 5.281 tA«i = — 12.445 — 5.300 4- 84.506 4- 6.185 4- 10.040 — 2.575 0..= — 8.191 — 16.377 4- 6.135 4-84.505 4-10.740 —8.214 --: — 9.371 — 6.G09 4- 10.040 4- 10.740 4- 5.672 — 8.316 ^ -f 7.951 4- 5.281 — 2.575 — 8.214 — 3.816 4- 2.675 As we cannot expect a satisfactory determination of at an<l ta^v from these observations, we disregard the last two eqiiatioiiN a// and then, solving the first three, we obtain «?, jtaA:, and — - in terms of ah- and .tacy, as follows : »> :^ — 4".36 4- 0.375 AT — 0.525 rj^ee taA: = 4- .02 — 0.216 at — 0.004 ta^« ^= — 1 .83 — 0.095 AT — 0.010 TAt« These values substituted in the equations (A) give Q^^V 22- 44*.38 4- 0.651 at 4- 0.432 ta^^ fi == — 1 22 46 .64 4- 0.684 4- 0.443 fi = — 1 22 46 .58 4- 0.685 4- 0.442 Q=^ — l 22 44 .34 4- 0.653 4- 0.432 LONOITXTDE. S41 the mean of which, giving the second and third double weight, is (A") fi = — 1* 22" 45'.86 + 0.674 att + 0.439 r dee The equations (B) become fl'= 5» 7- 26-.99 — 1.314 att — 0.116 r^ee fi'=6 7 27.03 — 1.314 —0.101 the mean of which is (B") ^1= 5» 7- 27'.01 — 1.314 att — 0.109 r^ee Now, if we assume the longitude of Konigsberg to be well determined, we have Q = ai'+ yr = — 1* 22- 0*.4 + vy which, with the equation (A''), gives vr = ^ 46«.46 + 0.674 ajt + 0.439 xA^e Hence, by (B ''), the true longitude of Washington is !»"== C— vy = 5» 8- 12'.47 — 1.988 ajt — 0.548 n^^ee If the longitude of Washington were also previously well estab- lished, this last equation would give us a condition for deter- mining the correction of the moon's parallax. Thus, if we adopt fi,''=5* 8"* 12'.34, which results from the U.S. Coast Survey Chronometric Expeditions of 1849, *50, *51, and '55, this equation gives = + 0.13 — 1.988 ATT — 0.548 rAec whence Ar == 4- 0".07 — 0.276 rAee The probable value of A^e, according to Bessel, is within dz 0.0001, so that the last term cannot here exceed 0".10. If, therefore, the above observations are reliable and the supposed longitudes exact, the probable correction of the parallax indi- cated scarcely exceeds 0".l, a quantity too small to be established by so small a number of observations. Nevertheless, the example proves both that the adopted parallax is very nearly perfect, and that a large number of observ-ations at various well determined places in the two hemispheres may furnish a good determination of the correction which it yet requires. 542 LUNAR ECLIPSES. Finally, the corrections of the Ephemeris in right ascension and declination, according to the above determination of y *^d i>, are found by (586) to be (putting a' for a and i' for d) A(o — o') = A(^ — ^') = — 28".42 + 0.469 Ar + 0.187 r^t^et — .48 + 0.314 ATT — 0.556 r^tLtt This large correction in right ascension agrees with the reenlu of the best meridian observations on and near the date of this eclipse. Since that time the Ephemerides have been greatly improved. LUNAR ECLIPSES. To find whether near a given opposition of the moon and Aa a lunar eclipse will occur. — The solution of this prob- lem is similar to that of Art. 287, except that for the sun*s semidiameter there must be substituted the apparent semidiameter of the earth*s shadow at the distance of the moon ; and also that the apparent distance of the centres of the moon and the shadow will not be affected by parallax, since when the earth's shadow falls upon the moou an eclipse occurs for all observers who have the moon above their horizon. If S, Fig. 45, is the sun's centre, E that of the earth, L3I the semidiameter of the earth's shadow at the moon, we have Apparent scmidiamotcr of the total shadow = LEM = BLE—EVL = BLE—{AES-^ EA V) where we employ the same notation as in Art. 287. But ob8er\'atiou has shown that the earth's atmosphere increases the apparent breadth of the shadow by about its one- fiftieth part:* so that we take * This fractional increase of the breadth of the shadow was giTen bj Lambibt m ^, and by Mater as J^. Beer and Madlbr found ^ from a number of obserrmlioM of eclip8es of lunar spots in the very favorable eclipse of December 2G, 1833. S«t ** Der Mond nach teinen koimiachen und individuellen Verhiltnisten, oder ali^rmfmt vtrgleichcnde Stlenographie^ von Wilhelm Beer und Dr. Joiuiflf HliXBlCR MJiOLKB,'* {OS. LUNAR ECLIPSES. 643 App. somid. of shadow = — (tt — «' + ^) (^^7) In order that a lunar eclipse may happen, we must have, therefore, instead of (477), P C08 7'< |1 (,r - «'+ ^) + 5 (688) 50 or, taking a mean value of i', as in Art. 287, /9< [1^ (^ - «'+ ^) +«] X 1.00472 Employing mean values in the small fractional part, we have [S ^"^ - ^+ ^) + ^1 >^ .00472 = 16" and the condition becomes /9<|^ (^-«'+ ^) + « + 16" (589) K in this we substitute the greatest values of ;r, tt', and s, and the least value of s'^ the limit /9 < 63' 53" is the greatest limit of the moon's latitude at the time of opposi- tion for which an eclipse can occur. K we substitute the least values of ;r, n'^ and s, and the greatest value of «', the limit /9<52'4" is the least limit for which an eclipse can fail to occur. Hence, a lunar eclipse is certain if at full moon ^ < 52' 4", impossible if ^ > 63' 63", and doubtful between these limits. The doubtful cases can be examined by (589), or still more exactly by (588), employing the actual values of ;r, ;r', 5, «', at the time, and computing /' by (475). ' These limits are for the total shadow. For the penumbra we have 51 App. semid. of penumbra = — (» + ^ + '^) (690) 544 LUNAB ECLIPSES. 80 that the condition (588) may be employed to determine whether any portion of the penumbra will pass over the mooiif by substituting + 5' for — «'. It will be worth while to make this examination only when it has been found that the total shadow does not fall upon the moon. 339. To find the time when a given phase of a lunar eclipse iriB occur. — The solution of this problem may be derived from the general formulie giren for solar eclipses, by interchanging the moon and earth and regarding the lunar eclipse as an eclipse of the sun seen from the moon ; but the following direct investigation is even more simple. Let Sy Fig. 46, be the point of the celestial sphere which is opposite the sun, or the appar- ent geocentric position of the centre of the earth's shadow; My the geocentric place of the centre of the moon ; P, the north pole. If we put a = the right ascension of the moon, a = the right ascension of the point S^ = R. A. of the sun + 180^, d = the declination of the moon, d^ = the declination of the sun, Q = the angle PSAf, L = SM, we have — d'= the declination of 5, and the triangle PS3I gives sin X sin Q= cos o sin (a — a') 1 /'«;qn sin L cos Q = cos d' sin d -\- sin ^' cos d cos (a — o') J The eclipse begins or ends when the arc SM\^ exactly equal to the sum of the apparent semidiameters of the moon and the shadow. The figure of the shadow will differ a little from a cift»le, as the earth is a si)heroi<l; but it will be sufficiently accu- rate to regard the earth as a sphere with a mean radius, or that for the latitude 45°. This is equivalent to substituting for r iu (587) and (590) the parallax reduced to the latitude 45°, which may be found by the formula LTJNAB ECLIPSES. 540 TT, = [9.99929] ^ (592) where the factor in brackets is given by its logarithm. Hence the first and last contacts of the moon with the pe- numbra occur when we have J^ = ^(.^ + ^+^) + s (593) For the first and last contacts with the total shadow, Z = ^(n^^s!+^ + s (594) For the first and second internal contacts with the penumbra, I^ = ^i^. + ^+0-s (595) For the first and second internal contacts with the total shadow, or the beginning and end of total eclipse, i = |l(;r,-5'+^-« (596) The solution of our problem consists in finding the time at which the equations (591) are satisfied when the proper value of L is substituted in them. A very precise computation would, however, be superfluous, as the contacts cannot be observed with accuracy, on account of the indefinite character of the outline both of the penumbra and of the total shadow. It will be suffi- cient to write for (591) the following approximate formulse, easily deduced from them : Zr sin § = (a — a') cos ^ \ sin 1" ■ Let ns put sin 2 ^ sin' 1 (a — a') *"" Binr a: = (a — a') cos a ) (598) a/, y'= the hourly increase of x and y , then, if the values of x and y are computed for several successive Vol. L— 86 546 LUNAB ECLIPSES. hours near the time of full moon, we shall also have r? and }f from their differences ; and if Xq ^^^ Vq denote the values of x and y for an assumed epoch Jij ^^^^ the time of opposition, we shall have for the required time of contact T= 2J + r the equations X sin § = rCg + a/r jCr cos C = y„ + y'^ from which r is obtained by the process already frequently employed in the preceding problems. Thus, computing the auxiliaries m, JET, n, N^ by the equations } (599) m%\n M= x^ fiBin N= of mQoaM=y^ ncosN=y' we shall have m sin (M — N) sm 4 = ^-i ^ L cos 4 m cos (jlf — N) \ (600) n n in which we take cos '4' with the negative sign for the first contact and with the positive sign for the last contact. The angle Q = N+ ^^z is very nearly the supplement of the angle PMS^ Fig. 46 ; from which we infer that the angle of posi- tion of the point of contact reckoned on the moon's limb from the north point of the limb towards the east = 180° + N+ '\i/. The time of greatest obscuration is found, as in Art. 324, to be which is also the middle of the eclipse. The least distance of the centres of the shadow and of the moon being denoted by J, we have, as in Art. 324, A = ±msin{M-^N) (602) the sign being taken so that J shall be positive. If then we put D = the magnitude of the eclipse, the moon's diameter being unity, LUNAR ECLIPSES. 647 we evidently have D = L — A 28 (603) in which the value of L for total shadow from (594) is to be employed. The small correction e in (598) may usually be omitted, but its value may be taken at once from the following table : Value of r. 6 «— a' 0" 1000" 2000" 8000" 4000" 6000" 6000" O" 0" 0" 0" 0" 0" 0" 0" 6 1 2 3 5 8 10 2 4 7 10 15 15 1 2 6 10 15 22 20 1 3 7 18 19 28 25 1 4 8 15 28 33 30 1 4 9 17 26 38 The quantity c has the same sign as ^, and is to be subtracted algebraically from d -\- d\ Example. — Compute the lunar eclipse of April 19, 1856. The Greenwich mean time of full moon is April 19, 21* 5'*.5. We therefore compute the co-ordinates z and y for the Greenwich times Apr'd 19, 18*, 21*, 24*. 18» 21* 24* 5 R.A. = a 0RA. + 180« = a' a a' a — a' (in arc) 5 Decl. -^fi li = if ■ c y log (a — a') log cos d log X 13* 46- 36-.62 13 52 52.98 — 6 16.36 — 5645" _llo2r 0".2 +11 35 49 .4 + 13 . + 542" n3.75166 9.99127 n8.74298 13*52- 9*.81 13 53 20.93 — 1 11.12 — 1067" — 12<> 6'43".7 +11 38' 22 .8 0. — 1701" n8.02816 9.99022 n3.01^38 13*57-45M2 13 53 48.88 + 8 56.24 + 8544" -12^46' 5".5 +11 40 56 .6 + 6. — 8903" 8.54949 9.98913 8.53862 548 LUNAB ECLIPSES. Hence we have the following table : 18* 21 24 X Diff. — 8** y Diff. = 8/ I — 5533" 1043 + 3456 4-4490 +4499 x'= + 1498 + 542" — 1701 — 8908 — 2248 — 2202 y'--741 To find L, we have, by (693) and (694), rr = 54' 32" r, = 8267" «'= 957 «'= 9 „j _ s* + ^ jV ('f. - «' + ^) « = 2319 46 891 L for shadow = 3256 X for penumbra = 5209 4283^ 85 891 Assuming the time TJ = 21*, we proceed by (599) and (600) : m sin M m cos Jtf" M log m 1048 1701 210^ 31'.0 3.3000 a/ = n sin JV if = n cos JV logn + 1498 — 741 116^ 19^.2 3.2230 7t% COS (M — JV) T.= + 0*.108 21 iTj = Time of middle of eclipse = 21 .108 Shadow. Penumbra. log sin -4^ L cos 4 n 9.7855 HP 1*.542 21 .108 9.5815 qi 2».881 21 .108 Beginning End 19 .566 22 .650 18 .227 23 .989 For the magnitude of the eclipse, we have, by (602) and (603) : OCCULTATIONS OP FIXED STARS. 549 m Bin {M — If) = IP,: = J = : 1987" L = :3256 L- - A — :1269 2a = = 1782 i> = l?l^ = 0.71 1782 For the position of the points of contact with the shadow, we have, from the above value of log sin 4^ for shadow, taking cos 4^ as negative for first and positive for second contact, lit Contact. 2d Contact. N 180»H-JV+ + 142<» 24' 116 19 78 43 37" 36' 116 19 833 55 and hence 1st contact is 79® from north point of limb towards the east, 2d 26® *< " *< " « west. The times of the several contacts for any meridian are obtained from the times above found by subtracting the west longitude of that meridian. OCCULTATIONS OF FIXED STARS. 840. The occultation of a fixed star by the moon may be treated as a simple case of a solar eclipse, in which the sun is removed to so great a distance that its parallax and semidiameter may be put equal to zero. The cone of shadow then becomes a cylinder, and the point Z of Art. 289 is nothing more than the position of the star, so that the co-ordinates of the moon at any time are found by the formulae (482) by regarding a and d as the right ascension and declination of the star. In like manner the co-ordinates of the place of observation will be found by (483). The radius of the shadow is constant and equal to A, which is, therefore, to be substituted for L = I — i^ in (490) and (491). The co-ordinates z and ^ will not be required unless we compute the latter for the purpose of taking into account the efiect of refraction according to Art. 327. For the convenience of the computer I shall here recapitulate the formulse required in the practical applications, making the modifications just indicated. 550 OCCULTATIONS OF FIXED STABS. 341. To find the longitude from an observed occidtatum of a star b^ the moon. — According to the method of Art. 329, we proceed as follows : I. Find, approximately, the time of conjunction of the moon and star in right ascension, reckoned at the first meridian. Take from the Ephemeris, for four consecutive integral hours, two preceding and two following the time of conjunction, the moon's right ascension (a), declination (d), and horizontal parallax (r). Take also from the most reliable source the star's right ascensiou (a') and declination (d'). For each of these hours compute the co-ordinates x and y by the formula cos d Sm fa — a ) x= ^^ gin:r _ sin (^ — ^0 cos* i (tt -~ ttp + sin (d + d') sin' j (> — >^ sin K and, arranging their values in a table, deduce their hourly variations x^ and y' for the same instants for which x and y have been computed. n. Let ji be the local sidereal time of an observed immersion or emersion of the star at a place whose latitude is ^, and west longitude w ; t the corresponding local mean time. The co-or- dinates of the place are to be computed by the formulae A sin B = p sin ^' A cos B = p cos ^' cos (^fi — o') S = P cos f' sin (ai — a') 7i = A sin(J5 — a') C = A co8(J5 — <J') When log f is small, add to logs f and rj the correction for refraction from the table on p. 517. m. Assume any convenient time Tq reckoned at the tir?t meridian, so near to t + w that x and y may be considered to vary proportionally with the time in the inter\'al / + ai — 7^. For the assumed time, take the values of x and y (denoting them by To and y©), and also those of x' and y', and compute the aux- iliaries m. My &c. by the formulee LONGITUDE. 551 m Bin Jf = ar« — f n sin iV = a/ '0 mcosM=y^ — ij ncoQlf=y' Bin + = niBiu(M-N) ^ _ ^^3^^^^^ k where i// is (in general) to be so taken that cos 4^ shall be nega- tive for immersion and positive for emersion (but in certain exceptional eases of rare occurrence, and of but little use in finding the longitude, see Art. 330). Then MccoB-^ km cos (M — N) n n or, when sin i// is not very small, hm sin (M — N — 4) n sin 4 K the local mean time / was observed, take h = 3600 in these formulse, and then the longitude will be found by But if the local sidereal time /i was observed, take h = 3609.856 in the preceding formulse ; then, /iq being the sidereal time at the first meridian corresponding to TJ, The longitude thus found will be affected by the errors of the Ephemeris. IV. To form the equations of condition for correcting the longitude for errors of the Ephemeris when the occultation has been observed at more than one place, compute the auxiliaries T,= T,- ^ {x,Bm N + y.cos N) » =: — x^ cos N -\- y^Bin N nn the same value of h being used as before. * According to Ovdbmahs {Astron. Naeh., Vol. LI., p. 80), we shoald ase for oocol- UtionB k = 0.27264, or log k = 9.485590, which amounts to taking the moon's apparent semidiameter about 1''.25 greater in occultations than in solar eclipses. Bat it is only for the reduction of isolated obserrations that we need an exact Talue, since, when we haye a number of obserrations, the correction of whateyer Talue of k we may use will be obtained by the solution of our equations of condition. 552 OCCULTATIONS OF FIXED STARS. Then, for each observation at each place, compate the coeffi- cients V tan '^j V sec 4^, and E=yn(t + lit ^ 2\) — «y tan 4 where o) is the approximate longitude and the unit of < + • — T, is one mean hour, and also F=ipn^n(t + w-T, ~»vtan4~Avsec4]~ '^^^'^^''^^^"'"^^ in which i? = f?^ log(l — ec)= 9.99709 Then, 0' denoting the true longitude, m'= m — vy* -J- V tan 4 . * + '^ sec 4 . itLk + ^ . ajt -f- J^ • ^^Aee in which f and tf have the signification Y = sin iV cos ^ A(a — a') + cos N a(^ — ^') * == — cosiV cos d A(a — a') + sin N a(^ — ^0 The discussion of the equations of condition thus formed may then be carried out precisely as in Art. 334, taking y*, 1?, ^ta*, at, and ntkce as the unknown quantities. Example. — The occultatiou of Aldebaraiij April 15, 1850, was observed at Cambridge, Mass., and Konigsberg, as follows:* At Cambridge, ^ = 42° 22' 48".6, a; = 4» 44- 30*. Immersion, 2* 1* 52*.45 Mean time. Emersion, 3 1 38.35 *' " At Konigsberg, ^ = 54° 42' 50".4, oi = — 1» 22- 0'.4 Immersion, 10* 57* 43v66 Sidereal time. Emersion, 11 47 47 .60 " " I. The Greenwich mean time of conjunction of the moon and star was about 7* 30*, and hence we take our data from the Nautical Almanac as follows : * AstroQoinioal Joarnal, Vol. I., pp. 189 and 175. LONOrCCDE. 553 I860 April 16. a 6 K 6» 65<> 66' 2rM6 + 16° 40' 0".05 58' 55".22 7 66 32 32 .06 16 46 30 .53 58 55 .87 8 67 8 46 .02 16 52 54 .77 58 56 .49 9 67 45 3 .02 16 59 12 .76 58 57 .10 The position of Aldebaran for the same date was a'=66°49'33".9 a'= + 16° 12' 1".7 Hence, by I. of the preceding article, we form the following table: Or.T. 6» X t! y y" 0.86519 + 0.58849 + 0.47664 + 0.10871 7 — 0.27671 47 .58581 63 8 + 0.81176 42 .69390 56 9 + 0.90014 82 .80243 48 n. The sidereal time of Greenwich Mean Noon, April 15, 1860, was 1* 33"* 8'.96. With this number, converting the Konigsberg times into mean times for the sake of uniformity, we find Ounbridge. Ktfnlgsberg. Immenkm. Emenion. ImmenioD. Emersion. 2* l"»62-.46 8* 1« 88'.86 9» 28«» 15'.64 10* 18« 1K88 6 46 22.45 7 46 8 .85 8 1 15.24 8 51 10.98 64<> 2' 2"M 69* 0' 58".86 164<> 25' 54". 90 1760 56' 54".00 347 12 28.66 2 11 24.45 97 86 21 .00 101 7 20 .10 9.826441 9.909898 9.869121 9.762689 n9.214324 8.451362 9.758801 9.735287 9.646065 9.641159 9.904088 9.922176 9.944427 9.952794 9.185091 8.549726 A* ;.~a' log p sin ^' log p cos ^' log^ log 7 logC The value of log C has been found in order to find the correc- tion for refraction. This correction is here quite sensible in the case of the Konigsberg observations which were made at a great 554 OCCULTATIONS OF FIXED STARS. zenith distance. By the table on p. 517, we find that the loga- rithms of f and 7] must be increased by 0.000006 for immersion, and by 0.000041 for emersion. Appljnng these corrections, the values of the co-ordinates are as follows : ^ 0.168S0 + 0.02827 4- 0.57886 + 0.54366 n -f 0.44266 + 0.48768 -f 0.80175 + 0.88602 lii. Assuming convenient times not far from t + w^ we have Assumed T^ 6*.8 7*.8 8».0 8*. 85 'o — 0.89440 + 0.19406 + 0.81176 + 0.81188 Vo + 0.6686S + 0.67218 -f 0.69890 + 0.78615 x^ — ^ = m sin if 0.28460 + 0.16579 — 0.26210 + 0.26822 y^ — tf m cos M 4- 0.12092 + 0.28460 — 0.10785 — 0.04987 M 297«> 40* 16".6 850 15' 86". 1 2470 88' l^.O 100» 81' 57-.7 logm 9.415608 9.458164 9.452488 9.485871 x* — n sin N + 0.68S47 + 0.58848 + 0.58842 + 0.58836 y* n cosJV + 0.10865 + 0. 10857 + 0.10856 + 0.10S49 N 79« 82* 21M 790 82* 45".8 790 82' 48^.5 79*>88' 8-.6 4 216 11 85.9 812 88 59 .0 167 85 28 .5 21 1 28 .1 (A — 8600) T — 89'. 74 — 128'.82 — 68'.68 a-.M u 4* 44"» 37'.81 4* 44« 12'.88 — 1*22^ 7'.01 1 1*22» 4'.90 iV. For the equations ot condition, taking T^ — 7*.8, r, — 7*.2772 It — 3536" and putting - + .6258 — the true 1( log )ngitudo of C E V — 0.2308 ambridge, i^onigsberg, we find, neglecting terms in A^e, w^ = 4» 44- 37'.81 —vr + 1-245 »^ — 2.108 rAA* — 1.293 at w. = 4 44 12 .83 — vr — 1.852 * + 2.515 taA' + 1.660 Ar 1 w iu. ;=. — 1 22 7 .01 — vT- — 0.374 .^ — 1.742 rAA- + 0.991 at »/= — 1 22 14 .90 — v/- + 0.654 * + 1.822 taA + 1.195 at whence the two equations = + 24'.98 + 3.097 d^ — 4.623 ttaA — 2.953 at = + 7 .89 — 1.028 * — 3.564 ittJi — 0.204 ajt If we determine (? and ;rAA: in terms of a^t, these equations give * = — 3".33 ^- 0.607 Ar ^taA = + 3 .17 — 0.232 Air LONQITUDB. 555 and then we find w^ = 4» 44- 26-.98 — vr — 0.048 a^ <w/= — 1 22 11 .29 — vr + 1169 A:r Assuming oi/ = — 1* 22~ 0*.4 as well determined, the last equa- tion gives vr = — 10'.89 + 1.169 A:r which substituted in the value of (o^ gives iu^ = 4» 44- 37'.87 — 1.217 att Finally, adopting the correction of the parallax for this date as given in Mr. Adams*s table (Appendix to the Nautical Almanac for 1856), Alt = -\- 5'M, this last value becomes «j = 4» 44- 31'.66 which agrees almost perfectly with the longitude of Cambridge found by the chronometric expeditions, which is 4* 44"* 31\95. With the same value of aj: we find r = — 2".90 * = — (K'.23 Ttiik = + 1".99 and hence, by (586), the corrections of the Ephemeris on this date, according to these observations, are A(a —a') = — 2".93 a(^ — a') = — 0".77 « The value niik = + 1''.99 gives aA: == 0.00056, and hence the corrected value k = 0.27227 + 0.00056 = 0.27283, which is not very different from Oudemans's value. (See p. 551). 342. When a number of occultations have been observed at a place for the determination of its longitude, it will usually be found that but few of the same occultations have been observed at other places. K, then, we were to depend altogether upon corresponding observations at other places, we should lose the greater part of our own. In order to employ all our data, we may in such case find for each date the corrections of the moon's place from meridian observations (see Art. 235), and, employing the corrected right ascension and declination in the computation of X and y, our equations of condition will involve only terms in TT^k and ^tt. The value of A;r will, however, be different on 556 0CCULTATI0N6 OF FIXED STARS. diflferent dates, and, therefore, if we wish to retain this term, we must introduce in its stead the correction of the mean parallax which is the constant of parallax in the lunar tables. K this constant is denoted by tt©, we shall have, very nearly, Aw = — AW- where t: is the parallax for the given date. The equations of condition will tihen be of the form where a = y8ec4 6 = — JP In Peirce's Lunar Tables, now employed in the construction of our Ephemeris, ;ro= 3422".06. 343. The passage of the moon through a well determined group of stars, such as the PleiadeSj affords a peculiarly favorable opportunity for determining the correction of the moon's semi- diameter as well as of the moon's relative place, of the relatiTe positions of the stars themselves, and also (if observations are made at distant but well determined places) of the parallax. Prof. Peirce has arranged the formulae of computation, with a view to this special application, for the use of the U. S. Coa^t Survey. See Proceedings of the American Association for the Adv. of Science, 9th meeting, p. 97. 344. When an isolated observation of either an immersion or an emersion is to be computed, with no corresponding obtiena- tions at other places, it wnll not be necessarj' to compute the values of x and y for a number of hours. It will be sufficient to compute them for the time t -\- (o (t being the obsen-ed local mean time, and o) the assumed longitude) ; and, as the corre<-tion of this time will always be small, the hourly changes may l>e found with sufficient precision by the approximate formula?, easily deduced from (482), xf= — cos ^ V=z PBEDICTION FOR A GIVEN PLACE. 557 where da and dd denote the hourly increase of a and d respect- ively, 345. To predict an occultation of a given star by the moon for a given place on the earth. — ^We here suppose that it is already known that the star is to be occulted at the given place on a certain date, and that we wish to determine approximately the time of immersion and emersion in order to be prepared to observe it. The limiting parallels of latitude between which the occultation can be observed will be determined in the next article. For a precise computation we proceed by Art. 322, making the modifications indicated in Art. 340. But, for a sufficient approximation in preparing for the obser- vation, the process may be abridged by assuming that the moon's right ascension and declination vary uniformly during the time of occultation, and neglecting the small variation of the parallax. It is then no longer necessary to compute the co-ordinates x and y directly for several different times at the first meridian, but only for any one assumed time, and then to deduce their values for any other time by means of their uniform changes. It will be most simple to find them for the time of true conjunction of the moon and star in right ascension, which is readily obtained by the aid of the hourly Ephemeris of the moon. Let this time be denoted by Tq. We have at this time x = 0, and the value of y will be found with sufficient accuracy by the formula n in which d, ;r, are the moon's declination and horizontal parallax at the time Ji, and 8^ is the star's declination. Let Aa (in seconds of arc) and a5 here denote the hourly changes of the moon's right ascension and declination for the time Tq. Then we have, nearly, A» / A^ of — — cos d y — — It ^ K Let 7\ be any assumed time (which, in a first approximation, may be the time Tq itself). Then the values of the co-ordinates at this time are 558 OCCULTATIONS OF FIXED STARS. and to find the time ( T) of contact of the star and the moon*s limb, we shall, according to Art. 322, have the following formulae: in which /Eij is the sidereal time at the first meridian corresponding to Tj, a' is the star's right ascension, and (o is the longitude • ABinB = pBin^ ( ^ p cos f'sin ^ A cos B = p cos f' cos * Tj = Asin(^B — ^') p! = 54148 sin 1" f = /*' A cos J5 log /i' = 9.41916 1?' = /*' e sin d' m sin M=x — f n sinN = af — f m cosJtf" = y — ly n cosJV= y — V Bm4 = ""'"^^~^ log* = 9.48600 k cos 4 mcos(3f — N) n n where '4' i^ to be taken so that cos '^ shall be negative for immersion and positive for emersion. For a second approximation, we take T as the assumed time Tj and repeat the computation for immersion and emersion separately. The new value of i> for this second approximation will be most readily found by adding the sidereal equivalent of T (converted into arc) to its former value. It is more especially desirable to know the true time of emersion, and the angle of position of the point of reappearance of the star. Since this angle in solar eclipses was reckoned on the sun's limb, while here it must be reckoned on the moon's, it will be equal to 180 + Q: so that, taking the value of i^ froni the last approximation, we shall have Angle of pi. of contact from the north pt. of the moon's limb J = 180^ + ivr+ 4 For the angle from the vertex of the moon's limb, we find ^ by the equations |) sin /^ = f + f 't pcoBr = Ti + v''^ PREDICTION FOR A GIVEN PLACE. 659 where $, jy, f, rf^ t are to be taken from the last approximation ; and then Angle of pt of contact fl^m | _ + jy^ + ^ _ the Yertex of the moon s limb j ■ i ^ / If the computation in any case gives m sin {M— N) > A:, we have the impossible value sin -^^ > 1, which shows that the star is not occulted at the given place. If we wish to know how far the star is from the moon's limb at the time of nearest approach, we have (Art. 324) J = rh wi sin (M — N) the sign being taken so that J shall be positive. This is the linear distance of the moon's centre from the line drawn from the place of observation to the star, and therefore the angular distance as seen from the earth is nJ. The apparent semidiameter of the moon is nk^ and hence the apparent distance of the star from the moon's limb is ;r(J— A).* Example. — ^Find the times of immersion and emersion in the occultation of Aldebaran^ April 15, 1850, at Cambridge, Mass. The elements of this occultation have been found on p. 558, with which an accurate computation may be made by the method of Art. 322 ; but, according to the preceding approximate method, we proceed as follows. The Greenwich time when the moon's right ascension was equal to that of the star is found, frt)m the values of a on p. 553, to be 2; = 7*.47 = 7* 28- 12*. For this time we have Att = 4- 2174'' ^ = + le** 49' 31".l A^ = + 384 d'z= 16 12 1 .7 ^= 3536 ^ — ^'=+ 2249" whence, by the above formulee, y^ = + 0.6360 a/= + 0.5886 2^= + 0.1086 Then the computation for Cambridge, tp = 42° 22' 49", 01 = 4* 44* 30*, will be as follows. For the first approximation, we assume 7\ = 7i, and hence we have * More exactly, allowing for the augmentation of the moon's semidiameter, it is IT (J — *) (1 -J- ^ sin tt), where we have ^ = ^ cos (^ — i'). 560 OCCULTATIONS OF FIXED STABS. T,= 7* 28- 12*. Sid. time 6r. noon = 1 33 9 .0 Eeduction for T^= 1 13 .6 M,= 9 2 84.6 a'= 4 27 18.3 la =: 4 44 SO ^ — a'— iii = * = 23 60 46.8 = 357* 4r.6 with which we find the following results : X = 0. f — — 0.0298 m sin Jtf" = + 0.0298 M= 8" 82'.4 3> = + 0.6886 f' = + 0.1940 n sinJV — 4- 0.8946 N— 74» 19'.1 log sin 4 = n9.8395 mcos(Jlf - n -■^)_ : — 0M690 For immersion. T = — 0» .6491 T, = 7 .4700 T — 6 .8209 T — 6» 49- 15' w — 4 44 30 Local time — 2 4 45 y = + 0.6360 7 = + 0.4377 m cos 3f = + 0.198S log m — 9.3021 }f- + 0.1086 V- — 0.0022 ncosJV = + 0.1108 log 11 = 9.6127 log cos 4 — 9.8590 A: cos 4 7 0^.4801 n For emenioB. r — + 0*.8111 ^1 = 7.4700 T = 7 .7811 T — 7» 46- SS* a 4 44 30 .ocal time — 3 2 22 These times are nearly correct enough ; but, for a more accurate time of emersion, we now assume T^= 7*.7811, with which we find a; = a/ (2; — 2;) = + 0.1831 ^(2;— T;\= +0.0338 y, = + 0.6360 y = -f 0.6698 and to find the new value of t> we have r = + 0*.8111 = 18*40*, the sidereal equivalent of which is 18* 43*. 1, or in arc 4° 40'.8. This, added to the above value of i?, gives the corrected value I? = 2° 22'.4. Repeating the computation with these new values of X, y, and i?, we find LIMITING PARALLELS. 561 r-oo^(^-J^ = _ 0..5082 * = I"' ^2' n N= 74 66 ^^^«* = + .4901 ^^' n • 212 17 T = — 0.0181 r= 3 33 2\= 7 .7811 208 34 !r== 7.7630 , ^, ^ xniooi»T/ 7»4>17« ( ^*^® ^**' reappears at 212® 17' T 1 x« o 1 1^ ifrom the north point, or 208® 34' Local time = 3 1 17 / « ., . ^ „\, jn'om the vertex, of the moons (limb. This time agrees within 2V with the actually observed time of emersion (given on p. 552). The principal part of the difference is due to the error of the Ephemeris on this date. 846. To find the limiting parallels of latitude en the earth for a given occuUation. — The limiting curves within which the occulta- tion of a given star is visible may be found by the general method given for solar eclipses, Art. 311, which, of course, may be much abridged in such an application. But, on account of the great number of stars which may be occulted, it is not pos- sible to make even this abridged computation for all of them. The extreme parallels of latitude are, however, found by very simple formulae, and may be used for each star. For a point on the limiting curve, the least value of J in Art. 824 is in a solar eclipse = i, but in an occultation it is = k. Hence we have, by (557), the condition dr m sin {M — i\r) = k or, restoring the values of m sin M= x — $, m cos J!f = y — tj^ (x — ^) cos N — (y — )y) sin iV = it ^ The angle iVis here determined by the equations (552); but, for an approximate determination of the limits quite sufficient for our present purpose, we may neglect the changes of f and jy, and take nsiniV=a/ ncosiV = y' Let Xq and y^ be the values of x and y for the assumed epoch Tq ; then for any time T= TJ + r we have X == rTg 4" ^ sin iV . r y = y^ -|- n cos JSf . t Vol. L— 36 — ■ •-■& .• — >, — T sin -V = — k - -i i liz-.a :i Tv* - 'w'-r Liivc. by negiectin^ the coin- er: j- = -r :-:s 3' — Z sin -5' .^ - _-~ T r V ttrcmiize the maximum and mininom values ^ ._ L^ Ilr^ JCC'iii-jELS. Let Us put f = — ^ cos y ^ r^ sin X > = c *in 3' -r 15 cos -Y c = — a cos -\' -p 6 sin JV 7 = a sin J\' -|- ^ cos -V : = ^ . 1 _ a' — 6«; "•Six «« *»<^ oa^^- bv our first condition, tz = — J, COS -y + y, sin -Y ±: it ^ , vi!>ni:i: ^-lanritT, since wc may here assume jr' and / >^ ^» *\ Jiav: r — :-— ^= 1, we can assume y and e so as to COS / = a sin ;* cos t = b sin 7 sin c = C J »:'.ii«i si-i r •-'^^^' ^'^^ restricted to positive vahies. The formula ^ , c^ r *iti -V cos J' + sin 7^ cos e cos Xcos d' + sin r sin e sin S' ^^.:t »d^j" K* put under a more simple form by assuming fi and j^ « CO ^dkCislY the conditions sin t^ = sin iV cos $' cos ,3 cos X = cos N cos d' cos j3 sin A == sin d' ^^kieA ^x^ i^ i^^^^y ^^ restricted to positive values. LIMITING PARALLELS. 563 We thus obtain sin f = sin /9 cos y -f cos /9 sin y COS {X — e) in which ip and e are the only variables. Since cos j9 sin y is positive, this value of sin ^ is a maximum when cos (X — e) = 1 or X — e = ; and a minimum when cos {X — e) = — 1, or X — e = 180°. Hence we have, for the limits, sin ^ = 8in(^ ± y)^ that is for the northern limit, f> = ^ -(- /* for the southern limit, 9 = P — y One of the points thus determined may, however, be upon that side of the earth which is farthest from the moon, since we have not restricted the sign of ^, and our general equations express the condition that the point of observation lies in a line drawn from the star tangent to the moon's limb, which line intersects the surface of the earth in two points, for one of which (^ is positive and for the other negative. But, taking ^ only with the positive sign, we must also have sin e positive. For the northern limit, therefore, when A = e, sin ^ must be positive, which, according to the equation cos ^ sin A = sin 5', can be the case only when 5' is positive. Hence the formula <p = fi + T gives the most northern limit of visibility only when the star is in north declination. For similar reasons, the formula <p = ^ — y gives the southern limit only when the star is in south declina- tion. The second limit of visibility in each case must evidently be one of the points in which the general northern or southern limiting curve meets the rising and setting limits, — namely, the points where ^ = 0, and consequently, also, sin e = 0, cos e = ± 1, which conditions reduce the general formula for sin ^ to the following : Bin f) = (sin N cos y dz cos iV sin y) cos ^' = sin (N ±: y) cos d' K cos iVis taken with the positive sign only, the upper sign in this equation will give the most northern limit to be used when the southern limit has been found by the formula <p = ^ — y, and the lower sign will give the southern limit to be used when the northern limit has been found by the formula <p = fi -\- y. Finally, since the epoch T^ is arbitrary, we may assume for it the time of true conjunction in right ascension when 0^)= 0, and we shall then have a = cos /' = y^ sin iV ± A' 564 OCCULTATIOKS OF FIXED STABS. The above discussion leads to the following simple arrangement of the formulfiB COS r^ = y„ sin N ±i 0.2723 (r < ISO**) sin ^ = sin i\r cos d' (/9 < 90*) 9, = fi±ri ) (604) cos rt = j/o sin iV =F 0.2723 Bin ^, = sin (iV rp r») cos ^' (iV< 90<*) in which the upper or the lower signs are to be used, according as the declination of the star is north or south. When the declination is north, ^^ will be the northern limit and y, the southern ; and the reverse when the declination is south. The angle Nis here supposed to be less than 90°, and is found by the formula taniV = ^ always considering y' as well as x' to be positive. When the cylindrical shadow extends beyond the earth, north or south, we shall obtain imaginary values for y^ or y^ The following obvious precepts must then be observed : 1st. When cos 7*1 is imaginary, the occultation is visible beyond the pole which is elevated above the principal plane of reference, and, therefore, we must put for the extreme limit <fi^^-r 90®, or <fi= — 90°, according to the sign of 5'. 2d. When cos y^ is imaginarj^ the value of ^^ will be the lati- tude of that point of the (great circle) intersection of the prin- cipal plane and the earth's surface which lies nearest the depressed pole; that is, we must take ^^=d'—90^, or f, — o'-!-90% according as o' is positive or negative. It is also to be observed that the numerical value of f| obtained by the formula <p^^= ^ ± y^ may exceed 90°, in which case the true value is either f^= 180° — (^ ±: ^,), or yj= — 180^ — (/3 =i= yi)y since these values have the same sine. Example. — Find the limiting parallels of latitude for the occultation of Aldebaran^ April 15, 1850. We have found, page 559, for this occultation, y^= + 0.6360 x' = 0.5886 ^ = 0.1086 Hence, with 3' == 16° 12', we find OCCULTATIONS OF PLANETS. 565 '7f N = 79^ 33' log sin /9 = 9.9751 y, sin iV = + 0.6255 p = 70^ 47 k= 0.2723 ri = 26 8 cos r, = + 0.8978 /9 + ^'j = 96 55 cos z', = + 0.3532 S^i = 83 5 r, = 69<' 19' N—U= 10 14 ?>,= 9 49 It is hardly necessary to observe that the occultation is not visible at all the places included between the extreme latitudes thus found, since the true limiting curves do not coincide with the parallels of latitude, but cut the meridians at various angles, as is illustrated by the southern limit in our diagram of a solar eclipse, p. 504. Unless a place is considerably within the assigned limits, it may, therefore, be necessary in many cases to make a special computation, by the method of Art. 345, to deter- mine whether the occultation can be observed. OCCULTATIONS OF PLANETS BY THE MOON. 847. If the disc of a planet were always a circle, and fully illuminated, its occultation by the moon might be computed by the general method used for solar eclipses by merely substituting the parallax and semidiameter of the planet for those of the sun ; and this is the method which has generally been prescribed by writers on this subject. But with the telescopes now in use, and especially with the aid of the electro-chronograph, it is possible to observe the instants of contact with the planet's limb to such a degree of accuracy that it appears to be worth while to take into account the true figure of the visible illuminated portion of the planet. Moreover, the investigation of this true figure possesses an intrinsic interest which justifies entering upon it here somewhat at length. In order to embrace at once all cases, I shall consider the planet as a spheroidal body which even when fully illumi- nated presents an elliptical outline, and when partially illumi- nated presents an outline composed of two ellipses, of which one is the boundary of the spheroid and the other is the limit of illumination on the side of the planet towards the observer. 1 begin with the determination of the first of these ellipses. 566 OCCULTATIOKS OF PLANETS. 348. To find the apparent form of (lie disc of a spheroidal planet* — Let us first express the apparent place of any point of the surface of the planet, by referring it to three planes perpen- dicular to each other, of which the plane of xy coincides with the plane of the planet's equator, while the axis of z coincides with the axis of rotation. In this system, let x,y,z = the co-ordinates of the point on the surface of the planet, f , iy, C = those of the observer. Straight lines drawn from the observer to the centre of the planet and to the point on its surface determine their apparent places on the celestial sphere. If these places are referred to the great circle which corresponds to the planet's equator, and if we put X, X' = the geocentric longitudes of the apparent places of the planet's centre and the point on its surface, reckoned from the axis of x, in the great circle of the planet's equator, /5,/9'=the latitudes of these places referred to the great circle of the planet's equator, p^ /o' = the distances of the centre of the planet and the point on its surface from the observer, we shall have (Arts. 32 and 33)t p cos fi cos X == — $ p cos /9 sin >l = — rj y (605) /o sin ^ = — C p' cos fi' cos X' = X — $ p' cos /?' sin i' = y — 13 )- (606) /o' sin ^ = 2 — C * The method of investigation here adopted, so far as relates to the apparent form of the disc, is chiefly derived from Bessel, Astronomische Unterauchungen^ Vol. I. Art. VI. f The group (606) may be deduced by supposing for a moment that the position of the observer is referred to a system of planes parallel to the first, but having its origin at the point on the surface of the planet. The co-ordinates in this system are equal to those in the first increased respectively by x, y, and z. The negative sign in the second members of both groups results from the consideration that the longi- tude of the observer as seen from the planet is ISO^-}- ^ ^^ 1S0^4~ ^'i ^^'^ ^^ latitude, — ^, or — ^'. Compare Art. 98. VORM OF A planet's DISC. 567 Now, let and C, Fig. 47, be the apparent Fig. 47. places of the planet's centre and the point on its surface, projected upon the celestial sphere; Q the pole of the planet's equator; Pthe pole of the earth's equator ; and let ^ = the apparent distance of C from = the arc OC, |/= the position angle of G reckoned at 0, from the declination circle OP towards the east, = POCy p= the position angle of the pole of the planet = POQ; then, in the triangle §0(7, we have sin ^ sin (j/ — P) = cos /9' sin (X' — X) sin s^ cos (p' — p) = cos fi sin /9' — sin /9 cos /9' cos (A' — X) Multiplying these by /t>', and substituting the expressions (605) and (606), we obtain p' sin s^ sin (p' — p) = p' sin «* cos (y — p) = — X BinX + y cos X — X sin /9 cos X — y sin /9 sin A -f ^ cos fi or, since s' is very small and /t>' sin 5' or /o's' differs insensibly jfrom josin 5' or ps\ ps' sin (p'—p) = — a: sin A 4- y cos A ) ps^ cos (y — 1>) = — X sin ^ cos X — y sin ^ sin il -|- ^ cos fi ) ^ ^ These equations apply to any point on the surface of the planet. K we apply them to those points in which the visual line of the observer is tangent to that surface, they will determine the curve which forms the apparent disc. The equation of an ellipsoid of revolution whose axes are a and 6, of which b is the axis of revolution, is (608) 1= — 4--^+- and the equation of a tangent line passing through the point whose co-ordinates are $, ij, and ^ is 1=^.1- y? + !? aa aa hb (609) The distances f, rj, and C are very great in comparison with x, 568 OCCULTATIONS OF PLAKST& y, and z. K we divide (609) by />, the quotients -? ^ - will be X v z X of the same order as -, -, ^, but the quotient - will be inappre- X V z ciable in relation to the quotients — ? -=^, -rr-- Performinfir this ^ aa aa ho ° division, therefore, and substituting the values off, gy, and Z from (605), we may write for the equation of the tangent line f. X cos p cos k , y cos a%uik , z sin 3 ,/»,^x = h ^ ^ j^ (610) J£ the curve ACBy Fig. 47, is referred to rectangular axes passing through the apparent centre of the planet, one of which is in the direction of the pole of the planet, and if u and V denote the co-ordinates of any point of the curve, so that u = 5' sin (/?' — p) V = ^ cos (y — p) the equations (607) and (610) will enable us to determine x, y, and z in terms of u and v. Putting bb , — = 1 — ee aa the three equations become pu = — X Bin X -\- y cos X pv = — (x cos ^ + y sin X) sm fi -{- z cos fi = (xcoB X -{- y sin X) (1 — ee) cob fi + z ^n fi from which we derive — X sin X -{- y COB X = pu , . , sin /9 — X cos X — y sin X = pv 1 — ee cos'^ fl — ee) cos 8 z = pv ^ ^ 1 — ee cos'/9 Substituting these values in (608) and putting a s = - = the greatest apparent semidiameter of the planet, c = T|/(l — ee cos'/5) FORM OF A planet's DISC. 569 we find 55 = titt + - (611) which is the equation of the outline of the planet as projected upon the celestial sphere, or upon a plane passed through the centre of the planet at right angles to the line of vision. It represents an ellipse whose axes are 25 and 25 |/(1 -— ee cos^/S), e being the eccentricity of the planet's meridians. The minor axis {OBy Fig. 47) lies in the direction of the great circle drawn to the pole of the planet's equator. We next proceed to determine what portion of this ellipse is illuminated and visible from the earth. 849. To find the apparent curve of illummation of a planefs surface. — If the sun be regarded as a point (which will produce no sensible error in this problem), the curve of illumination of the planet, as seen from the sun^ can be determined by conditions quite similar to those employed in the preceding problem ; for we have only to substitute the co-ordinates expressing the sun's position with reference to the planet, instead of those of the observer. If, therefore, we put A, B = the heliocentric longitude and latitude of the centre of the planet referred to the great circle of the planet's equator, the equation of the tangent line from the sun to the planet, being of the same form as (610), will be ^ xcobBcoqA , vcosJ^sinyl . zainB = h ^ j^ — (612) aa * aa ^ bb ^ ^ If each point which satisfies this condition be projected upon the celestial sphere by a line from the observer on the earth, and u and V again denote the co-ordinates of the projected curve, we have here, also, to satisfy the equations pu = — X sm X -}- y cos X | pv = — (xco&X -{- y sin X) sin iS + z cos fi j (y V in which X and ^ have the same signification as in the preceding article. The values of x, y, and Zy determined by the three equations (612), (613), being substituted in the equation of the ellipsoid, we obtain the relation between u and v, or the equation 670 OCCULTATIONS OF PLANETS. of the required curve of illumination as seen from the earth. In order to fecilitate the substitution, let us put a:, == — a: sin A -f y cos X y^ = X cos A 4" y sin i from which follow x= — Xi sin ^ + yi cos X yz= Xi cos A + y, sin X At the same time, let us introduce the auxiliaries fi^ and B^ dependent upon ^ and B by the assumed relations - cos A = cos B — cos j^j = cos B g Cr - sm a,= r- Bin fi -^ sm Bi= t muB go Gr (614) Then the three equations become a = X, cos By sin (-4 — A) -(- y^ cos B^ cos {A — ^) + I ^ ^^^ ^i fiU = Xi -gpv = ^ yi sin fii + i^ cos fi^ from which we derive Xi= pU Ny^ = — pu cos /5j cos Bi sin {A — X) — - gpvBinBx Nt z = — /ou sin /5iC08 ^,8in (^1 — ^)-{-Tgp^ cos J5iC0S(yl — X) where, for brevity, iVis put for sin ^^ sin B^ + cos^j cos B^ cos {A — >l). Before substituting these expressions in the equation of the ellipsoid, it will be well to consider the geometrical signification of the quantities /9i and By If we draw straight lines from the centre of the planet to the earth and to the sun, the latitudes of the points in which these lines intersect the surface of the planet will be /? and B. If these points be projected upon the surface of a sphere circumscribed about the ellipsoid, by perpendiculars to its equator, the latitudes of the projected points will be ^, and jBj ; and g and G will be the corresponding radii of the ellipsoid. If now these projected points are referred to the celestial sphere, by lines from the planet's centre, they will form with the pole Q of the planet's equator a spherical triangle QOSy in which the rORM OF A planet's DISC. 571 angle Q will he A — X; and the sides including this angle will be 90° — A = QOy 90° — B,= QS. Denoting the angle at by 1(7, and the side OS by V, we shall have cos F = sin ^1 sin Bi + cos fi^ cos Bi cos (A — X)^ sin V cos w = cos A sin J?, — sin fi^ cos J?^ cos (A — X) > (616) sin F sin w = cos j?^ sin (yl — ^) J in which V is very nearly the angular distance between the sun and the earth as seen from the planet. This triangle also gives sin J5j = cos V sin fi^ -(- sin V cos fi^ cos w cos B^ cos (A — X) = cos F cos ^j — sin F sin fi^ cos w? cos B^ sin (il — 'I) = sin F sin lo By these equations the above expressions for x^, y^, and ^ are reduced to cos F. x^ = /DM cos F cos V.yi = — pu sin F sin ti? cos ^^ — r S'/oy (cos F sin /9j + sin F cos ^^ cos u?) cos V'j'Z = — pu sin F sin w sin /9j Substituting these in (608), observing that zx + yy = x^x^^ + y^^/j, we have cos'F— = uu cos'F + I (u sin M? + T S'^ cos w) sin F cos /?i + r ^^ ^ob V sin ^^ I -|- I (m sin i^ + - l/y cos w) sin F sin fi^ — j- gv cos F cos ^j I Developing the squares in the second member, and putting s for -, and also b € = i/Cl — ee cos* /5) = — we shall find SB = I M cos w? — V I + I M sm M? + r j sec* F (616) i72 OCCULTATIONS OF PLANBTS. which is the required equation of the curve of illumination, as seen from the earth, projected upon the celestial sphere. It represents an ellipse whose centre is at the origin but whose axes are, in general, inclined to the axes of co-ordinates, and, consequently, to the axes of the ellipse of equation (611). The equation (611) is only the particular case of (616) which corre- sponds to V= 0, or the case of full illumination. Fig. 48. 350. We have yet to determine what portions of the apparent disc are bounded by the two curves respectively. K ABA'B^, Fig. 48, is the ellipse of (611), which I shall call the first ellipse, and CDC'D' that of (616), which I shall distinguish as the second ellipse, the visible outline of the planet is composed of one- half the first and one-half the second curve, and these halves either begin or end at the points C and C, which are the common points of tangency of the two curves. These points satisfy both equations ; and, therefore, putting Wj and 1?! for the co-ordinates of either point, and subtracting (611) from (616), we find A / • . cos tr W , T^ = I M, sm t^ + Uj j tan' V which is satisfied, in general, by taking cos w ^ w, sm u? + r, = Denoting the position angle corresponding to u^ i\, by p^, we have 2/i= 5i 8in(pi — 7?), Vi=^ Sy co^^pi—p). Substituting these values, and also putting c, sm ic, = sin w c^ cos tr, = cos w (617) the preceding condition becomes whence Cj s^ cos (^j — p — tTj) = p^ = p + ir, :;: 90« (618) which expresses the position angles of both Cand C K we draw the arc 0D0% Fig. 48, making the angle BO(y = w?, and FORM OP A planet's DISC. 673 take 00' = Vj the point 0' will be nearly the position of the planet as seen from the sun, and the arc V will be the measure of the angular distance between the sun and the earth as viewed from the planet. If we assume sin w to be positive in equations (615), as we are at liberty to do, the arc 'Fwill be reckoned from the planet eastward from 0° to 860°. Now, so long asFis less than 180°, the west limb will evidently be the full limb, and when y is greater than 180°, the east limb will be the full limb. Hence we infer that a point whose given position angle is p' is on the east limb when P'>P + ^1 — 90° and < i? + w^ + 90° but on the west limb when P' < i? + w?i — 90° and > p -f t/?^ + 90° When V > 90° and < 270°, the planet is crescent ; but when 'F> 270° and < 90°, it is gibbous. In the case of a crescent planet there are two points, one on the full and the other on the crescent limb, corresponding to the same position angle : hence in observations of a crescent planet the point of observation on the limb will not be sufficiently determined by the position angle alone ; it will be necessary for the observer to distinguish the crescent from the full limb in his record. 351. In order to apply the preceding theory, it is necessary to find the quantities /), /9, >l, -B, A, Tlie direction of the axis of x in Art. 348 was left indeterminate, and may be assumed at pleasure, but it is most convenient to let it pass through the ascending node of the planet's equator on the equinoctial, so that X and // will be reckoned from this node. The position of the node must, therefore, be kno^vn, and this we derive from the researches of physical astronomers. If we put n = the longitude of the ascending node of the planet's equator on the equinoctial, i = the inclination of the planet's equator to the equi- noctial, we have at any given time ^, for the planets Jupiter and Saturn, the only ones whose figures are sensibly spheroidal. 574 OCCULTATIONS OF PLANETS. ^ (n = 367° 56' 25" + 3".59 (t — 1850) J^or jupiter. | . ^ ^50 25' 49" + 0".66(f - 1850) „ ^ f n = 125° 13' 54" + 128".76 (/ — 1850) + 0".0605 (t — 1850)« j^or&aiurn.| ,^ yoio'lO"— 15".08(f— 1850) + 0".0035(^— 1850)» in which t is expressed in years.* The values for Saturn apply either to its equator or the rings, which are sensibly in the same plane. If now we put o', d'z=z the right ascension and declination of the planet, we can convert a' and 8' into X and fi by Art. 23 ; we shall merely have to substitute in (29) or (31) a' — n for a, d' for 8, and i for e. The angle p is here the position angle of the pole of the planet reckoned from the declination circle of the planet towards the east; but in Art. 25 the angle tj is tlie position angle reckoned towards the west, and, therefore, we shall have to put 7j = 360° — p in (33). Hence we obtain the following formula for ^, iy and p : fBinF= tan ^' /' sin X = cos (F — i) f cosF = sin (a' — n) /' cos k = cos F cot (a' — n) tan ^ = sin A tan (jP — i) K (619) sin F' cot (o' — n) tan F'= tan i sin (o' — n) tan p = — QOB^F'—d') To find A and By we avail ourselves of the heliocentric longi- tude and latitude of the planets given in the British Almanac, and as these quantities are referred to the ecliptic, while A and B are referred to the planet's equator, we must know the rela- tive position of these circles. Putting N*^ the longitude of the node of the planet's equator on the ecliptic, 7'= the inclination of the planet's equator to the ecliptic^ N = the arc of the planet's equator between the equi- noctial and the ecliptic, * These values I have deduced from the data given in Damoi8EAU*8 Tablet icUp- tiquea dea Satellites de Jupiter^ Paris, 18^6 ; and Bessel's Bentimmung der Lagt und Grbtae dea Satuma-Ringea und der Figur und Groaae dea Satuma^ Aatronom, Xach.^ Vol. XII. p. 167. FORM OP A planet's DISC. 575 we deduce fix)m the data of Bessel and Damoiseau, for a given year <, r N'=SSb^ 40' 46"+ 49".80 (t — 1850) PorJupiterJ /'= 2<» 8'51"+ 0".43(^— 1850) {n = 336^ 33' 1 8"+ 46".55 (t — 1850) {N'= 167^ 31' 52"+ 46".62 (^ — 1850) /'= 28^ 10' 27"— 0".35(f— 1850) N = 43^ 31' 34"— 86".75 (t — 1850) — 0".0625(< — 1850)« and these values for Saturn also apply to the rings. Finally, if we put A', B'=the heliocentric longitude and latitude of the planet, referred to the ecliptic, the formulse (29) or (31) will serve to convert A' — iV' and 5' into A — N and B ; and they become X sin M= tan B' K' sin {A^N) = cos {M — J') JT cos Jf = sin (J'— iV'') JC'cosCyl— iVr)==cos Jf cot(il'— iV^') ( (620) tan jB = sin (il — N^ tan {M — 7') 352. The preceding complete theory admits of several abridg- ments in its application to the different planets, varying according to the features peculiar to each. Jupiter. — The inclination of Jupiter's equator to the ecliptic is 80 small that the quantity c = i/(l — ee cos* ^) never differs sensibly from |/(1 — ee), which, according to Struve's measures, is 0.92723. I shall, therefore, use as a constant the value log c = 9.9672. Again, on account of the small inclinations both of Jupiter's equator and of his orbit to the ecliptic, the angle w never differs much from 90°, and, since this angle is required only in computing the gibbosity of the planet (which never exceeds 0".5), it is plain that we may take w = 90°, and that V rasLj be found with sufficient accuracy by the formula or, indeed, by the formula F=il' — ;' (621) in which A' and A' are, respectively, the heliocentric and geo- centric longitudes of the planet, the former being taken directly 576 OCCULTATIONS OF PLANETS. from the British Almanac, and the latter computed from the geo- centric right ascension and declination by Art. 23 : so that for this planet the equations (615), (619), (620) will be dispensed with, except only the last two equations of (619), which will be required in finding 'p. Saturn. — The inclination of Saturn's equator to the ecliptic is over 28°, and therefore the quantity c = v (1 — ^^ cos*^) will have sensibly different values at different times. The value of — ^ is, however, given in the table for Saturn's Ring in our Ephemerides (where it is usually denoted by l). The value of et is 0.1865, or log ee = 9.2706. The gibbosity of Saturn is alto- gether insensible ; so that we shall have occasion to use only the equation (611), or in any formula that may be derived from the more general equation (616) we shall have to putF= 0. The angle p is also given in the table for the ring. Saturn's Ring. — The ring may be here regarded as an ellipsoid of revolution whose minor axis = 0. Hence we have only to make e = 1 in our formulae to obtain the equation of its elliptical outline. This gives c = i/(l — cos*^) = sin ^, which value being substituted in (611), we have at once the required equation, while the position of the ellipse is given at once by the angle p from the table above referred to. Mars, Veniis, and Mercury. — These planets may be regarded as spherical in the computation of their occultations, and we shall, therefore, have to consider only their crescent and gibbous phases. To adapt our formulae to the case of a spherical body, we have only to put c = 0, or c = 1. Since in this case we are concerned only with the apjmrcni figure of a partially illuminated spherical body, we may, for the convenience of computation, assume any point as the pole of the planet ; and it will be most natural to assume the point which is the pole of the great circle whose plane passes through the sun, the earth, and the planet p. ^g The direction of this pole is evidently the same as that of the line joining the cusps of the partially illuminated disc. This makes ^ == 0, ^ = 0, in (615), and, consequently, F= A — X. But, as the adopted equator of the planet is here a variable plane, we can no longer use the form (620) for finding A. A very simple and direct process for finding V offers itself. Let E, S, 0, Fig. 49, repre- PORM OP A planet's DISC. 577 sent the centres of the earth, the sun, and the planet; S'0'0", the great circle of the celestial sphere whose plane passes through the three bodies ; S' and (y, the geocentric places of the sun and the planet ; 0", the heliocentric place of the planet. Then Cy 0" is the arc heretofore denoted by V, and, in the infinite sphere, is the measure of the angle OOO"' = SOK Putting then V=0'0", Y = S'Cy^ and also R' = SO = the heliocentric distance of the planet, R =8E= « " « earth, we have Sin F = — sin y We might find V directly from the three known sides of the triangle SOE; but, as we have yet to find p, and y comes out at the same time with |) in a very simple manner, it will be prefer- able to employ the above form. To find p and 7-, let S', 0", 0", Fig. 50, be the three places above referred to, and P the pole of the equinoctial. Draw CyQ perpen- dicular to the great circle S^(yO''. This perpendicular passes through the adopted pole of the planet, and we have P<yQ=p, or PO'S'= 90° — p, and S'(y=y. Hence, denoting by 5' g, and D the declination of the planet and the sun, and by a' and A their right ascensions respectively, the spherical triangle PS'O' gives Y = sin d* sin D + cos 5' cos D cos (»' — A) \ p = cos d' sin D — sin d' cos D cos (a' — A) \ (622) cos y = sin y sin p sin y cos p = cos D sin (a' — A) } Hence, introducing an auxiliary to facilitate the computation, both p and V will be found by the following formulse : tan F = tan D sec (a' — A) tan p = cot (o' — A) sin (F — d') sec JP . -- R sin (a' — ^) cos D ( (^^^) sm V = -— ^^ ^ R' C08p In this method of finding V we do not determine whether it is Vol. L— 37 578 OCCULTATIONS OP PLANETS. greater or less than 90°. This is of no importance in computing an actual observation, but only in predicting the phase of the planet, whether crescent or gibbous. For the latter purpose we must have recourse to the triangle SUO of Fig. 49, the three sides of which are given in the Ephemerb. The value of V being found, the equation (616) will be used to determine the apparent outline after substituting c = 1 and w = 90°, whereby it becomes The value of s in our equations is supposed to be given. It will be most convenient to deduce it from the apparent semi- diameter of the planet when at a distance from the earth equal to the earth's mean distance from the sun, which is the unit employed in expressing their geocentric distances in the Ephe- meris. Thus, denoting the mean semidiameter by s^ and the geocentric distance by r', we have (Art. 128) «-i» ^-f' (624) and Sq may be taken from the following table : Mercury Venus Mars Jupiter Saturn Saturn's Rings Outer semi-major axis of outer ring Inner «* " ** «' Outer ** " inner " Inner " *« " «« h 8" .84 8 .66 5 .06 99 .70 81 .86 187 .56 165 .07 IGl .27 124 .75 Aathori^. Lb Verrier, Theory ofMerewnf. Peircs, Am. Ephemeru. (« (« «4 Struvb, A$tT. Naeh., No. 139. Bessel, Attr. Nach., No. 276. (( <( <( f Struve, Attr, A'acA.jNo. 139, reduced to agree with Bes- 8EL*s measures of the outer diameter of the outer ring. 353. To Jiiid the longitude of a i^lace from the obsei*ved contact of the moon's limb with the limb of a planet. — In the following investi- gation, it is assumed that the quantities p, m?, V^ c, are known for the time of the oceultation. They may be computed by the above methods for the time of conjunction of the moon and planet, and regarded as constant for the same oceultation over the e^rth in general. LONGITUDE. 579 Let 0, Fig. 51, be the apparent centre of the planet, and G the point of contact of its limb with that of the moon. Let OJ!f be drawn from towards the moon's centre, in- tersecting the moon's limb in D. Since the apparent seraidiameter of any of the planets is never greater than 31", it is evident that no appreciable error can result from our assuming that the small portion CD of the moon's limb coincides sensibly with the common tangent to the two bodies drawn at C K, then, the planet were a spherical body with the radius OD, the observed time of contact would not be changed. We may, therefore, reduce the occultation of a planet to the general case of eclipse of one spherical body by another, by substitutmg the perpen- dicular OD for the radius of the disc of the eclipsed body. Let s" denote this perpendicular; let OA and OQ be the axes of u and V respectively, to which the curve of illumination is referred by the equation (616) ; and let i? be the angle QOD which the perpendicular s" makes with the axis of r. The equation of the tangent line CD referred to these axes is u sin ^ -}- t; cos * = «" "We have also in the curve dv T- = — tan iJ du Differentiating the equation (616), therefore, we have (v sin M7 \ / . tan y% sin w \ u cos w 1 1 cos w -\ I (625) , / . . t?co8M?\/ . tan»9co8tr\ ._- ^ -f I ti sm w -\ M ®*^ ^ I ®®^ F = By means of this equation, together with (616) and (625), we can eliminate u and i?, and thus obtain the relation between 8 and 5". To abbreviate, put V sin w X = u cos w — y = u sin u? -[- c V COHW 580 OCCULTATIONS OF PLANETS. and also c' Bin d' = — c' cos i>' = cos * (626) then the three equations become X cos (d' — w) — y sin (v>' — w) sec«F = x«+y«sec«F=«« a: sin (i^' — tr) + y cos (v>' — u?) := — ^ From the first and second of these we find s sin (y — w) X = y = |/ [1 — co8» (i*' — \D) sin« F] 5 COS (*' — XD) COS* F l/[l — cos»(d'— M7) 8in»F] which substituted in the third give 5" = 5CC V[l — cos' (v>' — M?) sin'F] Hence, if we put sin / == cos (»>' — u?) sin F we have \ (627) 5" = 5.CC'C08/ 1 TVo have seen (Art. 352) that in all practical cases we may take w = 90°, and, therefore, instead of (626) and (627) we may employ the following : „, tan »J tan »^' = c sin / = sin t>' sin F ) (628) s sin f^ cos / 5"== sin y If the occultation of a cusp of Venus or Mercury is observed, we have at once 5"-- s cos «? (for the axis of v coincides with the line joining the cusps), and we do not require F. The value of s" is to be substituted in (486) for the apparent semidiameter of the eclipsed body. In that formula, ^denotes the apparent semidiameter at the distance unity : therefore, we must now substitute the value sin jy == r' sin «" LONGITUDE. 581 or, by (624) and (628), . „ sin 5. sin * cos y .^^. sin JU = ®— r — (629) sin d' ^ ^ Since / is here very small, we may put tan / = sin /, and the formula for L (488) becomes i = (2: — C)8in/± A: . .. sin J? . , , ^^ k sin 7r„ Hence, putting A' = A + (^-C)^4^ (630) ^^ we have i = (z - C) 5^^ zh ^ (631) rg When the angle i> is known, therefore, the preceding formulae will determine i, with which the computation will be carried out in precisely the same form as in the case of a solar eclipse, Art. 829. To find i?, let OP, Fig. 51, be drawn in the direction of the pole of the equinoctial ; then we have POQ = p, and, denoting POM by §, and Q has here the same signification as in the general equations (567), as shown in Art. 295 : so that when N and 4' have been found by (568) and (569), we have Q = N+'^, or ^ = N+ 4— p (632) But to compute ^^ by (569) we must know i, and this involves -ff, which depends upon i?. The problem can, therefore, be solved only by successive approximations; but this is a very slight objection in the present case, since the only formulae to be repeated are those for L and ^'j and the second approximation will mostly be final. It can only be in a case such as the occul- tation of Saturn's ring, where the outline of the eclipsed body is very elliptical, and especially when the contact occurs near the northern or southern limb of the moon, that it may be necessary (for extreme accuracy) to compute H a second time and, conse- quently, '^ a third time. The formula (629) is adapted to the general case of an ellip- 582 OCCULTATIONS OF PLANETS. soidal body partially illuminated, the point of contact being on the defective limb. When the point of contact is on the full limb, we have only to put V= 0, and the formula becomes .. jT sin 3^8in i» 8mg= (688) sm V and for the full limb of a spherical planet (Venus, Mercury, and Mars) we have H=^ s^. In the first approximation we may take L^= ± L 354. Sometimes it may not be known from the record of the obser\'^ation whether the point of contact is on the full or the defective limb of the planet. This might be determined by the method of Art. 350 ; but, since that method supposes the position angle p' to be given, which we do not here employ, the following more direct and simple process may be used. In that article the common point of tangency of the two curves of the full and defective limbs was determined by the condition cos to ^ u. sm w + V. = in which u^ and i\ denotes the co-ordinates of the point of tan- gency. Li the notation of Art. 353 this is simply y^ = ; and since we have 8 cos (»^, — w) cos' V ^' ~ ^/[l — co8»(.?^ — w) sin« F] it follows that we must have cos (f?j — M7) = or t^j == M? qi 90° ITcnce, when, as in our present application, we take w = 90®, we have \ = or ^^ = 180° Hence a point is to be regarded as on the east Umbfor values of d between 0° and 180°, and on tlie west limb for values of (? between 180° and 360° ; and (Art. 350) the east or the west limb is defective accord- ing as Vis between 0° and 180° or between 180° and 360°. But, since sin «?' and sin t? have the same sign, we deduce from this a still more simple rule ; for we have sin ;f = sin i?' sin F, whence it follows that the observed point is on the defective limb when sin ^ is positive^ and on the full limb when sin j[ is negative. LONGITUDE. 583 855. In the cases of the planets Neptune, Uranus, and the asteroids, the oecultation of their centres will be observed, and it will be most convenient to compute by the method for a fixed star, only substituting for tt the difference of the moon's and planet's horizontal parallaxes — ^that is, the relative parallax — in the formulse for x and y, Art. 341. This artifice of using the relative parallax may also be used with advantage for Jupiter and Saturn. Having thus found x and ^ as for a fixed star, we shall have, in the preceding method, X = (2 — C) ^^ d: * (634) the other formulae remaining unchanged. Example 1. — Several occultations of Saturn's Ring were ob- served by Dr. E^ane at Van Rensselaer Harbor on the northwest coast of Greenland during the second Qrinnell Expedition in search of Sir John Franklin.* The first of these was as follows : 1853 December 1